3?! 131.5} 19...: v t: .1. 3.1 3... ..w . 2 :3 A 3 . 1;: 3 fix. V. F»... T H.523?! I... 9.1.1). . )1 2.. 3 .25 . V , i it. x: . . ”anmi. 1.1.: i r... 4.1!!!! ¢ 5.91.233. illiiuv. .1. y). . a! ..q I1 «4.. . :) :5) I1; . THESIS 300‘ LIBRARY Pig’aichi‘gan Sisal-:2 University Om This is to certify that the dissertation entitled Synplect ic Approximat ion Of Hamiltonian Flows And Accurate Simulation Of Fringe Fields Effects presented by BELA ERDELYI has been accepted towards fulfillment of the requirements for PI-I D. degree in Physics ({4ka M Major professor M. Berz Date 3! 11101 M5 U is an Affirmative Action/Equal Opportunity Institution 0- 12771 PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRCIDateDuepes-nts SYMPLECTIC APPROXIMATION OF HAMILTONIAN FLOWS AND ACCURATE SIMULATION OF FRINGE FIELD EFFECTS B y Béla Erdélyi AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department. of Physics and Astronomy 2001 Professor Martin Berz ABSTRACT SYMPLECTIC APPROXIMATION OF HAMILTONIAN FLOWS AND ACCURATE SIMULATION OF FRINGE FIELD EFFECTS By Béla Erdélyi In the field of accelerator physics, the motion of particles in the electromagnetic fields of periodic accelerators is usually approximated by the iteration of a symplectic map, which represents the system over short time, such as one turn around the accelerator. Unfortunately, due to the complexity of the systems, in practice only some approximation of the one-turn map can be computed, as, for example, the truncated Taylor series. To this end, simulation of the nonlinear dynamics consists, in general, of the following three steps: 1) Computation of the truncated Taylor approximation of the one-turn map, 2) Symplectification of the Taylor map, and 3) Iteration of the resulting exactly symplectic map. This dissertation addresses all three components of the process, with the emphasis being on developing new methods that allow long-term tracking as accurately and efficiently as possible. Specifically, the contributions to the first step concern the fringe field effects. The truncated Taylor map should include every relevant effect, so that it is an accurate representation of the system over one turn. While it is straightforward to compute the truncated maps over the regions where the fields are independent of the longitudinal variable, it is not so anymore at the ends of the magnets, the so-called fringe field regions. We study fringe fields generically, to show their importance, and develop a method that allows “exact” fringe field map computation of superconducting magnets, for which the coils and the iron parts are represented by current wires. The theory is illustrated by a detailed study of fringe field effects on the nonlinear dynamics of the Large Hadron Collider at collision energy. Many contributions are established to the second step. It is well known that the theory of generating functions of canonical transformations provides a possible symplectification method. It is shown that, by transforming the dynamical problem into a problem in symplectic geometry, a general theory can be developed, which leads to a set of infinitely many new types of generating functions. It follows that it is possible to use this extended set to produce symplectic maps, and to reduce the whole set of generators to classes that give the same symplectified map. Moreover, the effects of factorization of the linear parts on the outcome of symplectification were studied. A variety of examples show the performance of various generator types, from which it can be concluded that it is not only important to symplectify, but also to symplectify “the right way”. The precise meaning of the last statement is the subject of the optimal symplectification theory, which can be formulated using methods of symplectic topology. In particular, Hofer’s metric allows the formulation of the optimality condition in a very general setting, and the solution leads to a generating function type (EXPO) that, in general, gives optimal results. In the proof, an interesting one-to—one correspondence between fixed points of symplectic maps and critical points of generating functions is developed, and a generalized Hamilton-Jacobi equation is derived. Finally, as contribution to the third step, it is pointed out that the numerical method used to solve the implicit equations arising in the iteration of the symplectic maps makes a difference in the final results, and, in general, a fixed point iteration is more robust than the widely used Newton method. Copyright by Béla Erdélyi 2001 To my family ACKNOWLEDGMENTS I gratefully acknowledge the support of many for their help and encouragement. I hOpe that every person who has helped me can take pride in the completion of this dissertation. I thank my advisor, Martin Berz, for providing all the support and resources necessary from start to finishing the thesis. His combination of direction and freedom provided an excellent training in scientific research, and contributed enormously to my professional growth. I learned a lot from him, and his enthusiasm, insight and commitment to science in general has been a great source of encouragement. I am also grateful for his investment in sending me to conferences. Many thanks for my guidance committee members: Jerry Nolen, Brad Sherrill, Bhanu Mahanti, and Dan Stump. Also, I would like to thank the physics graduate secretaries for the prompt and friendly help. I have benefited greatly from discussions with many other scientists: Carol John- stone, Weishi Wan, Francois Meot, Frank Zimmermann, Bill Ng, Jim Holt, Teddy Draghici, Frederick Siburg, and others. It has been a pleasure to work with the other members of the Beam Theory and Dynamical Systems group, who were at MSU over various periods of time be- tween 1996—2001: Vladimir Balandin, Lars Diening, Michael Lindemann, Jens von vi v-cw In“ ' n L-L.Lh¢.n ‘ O 4... 9- Z 7" Q? ad;.~.. ... Bergmann, Khodr Shamseddine, Kyoko Makino, Jens Hoefkens, and Shashikant Mani- konda. Jens Hoefkens deserves special thanks for many computer related help. Finally, I thank my family for their unconditional support, and especially my wife, daughter, mother, and sister whose love is a huge source of strength and great joy. vii Contel llSl OF I. llSI 0f fl 1 Sympl 1 lfllmdlli‘ 3 General lions l‘. Ti F 3‘; T'_ .- ) a ‘31: \“v :‘1 T 'J._ 1- L I) ‘J ‘1: y. 4‘. r A. 3 .J . Contents LIST OF TABLES xi LIST OF FIGURES xiii I Symplectic Approximation of Hamiltonian Flows 1 1 Introduction 2 2 General Theory of Generating Functions of Canonical Transforma- tions 8 2.1 The Local Theory ............................. 9 2.2 The Global Theory ............................ 17 2.2.1 Fundamentals of Symplectic Geometry ............. 18 2.2.2 Primitive Function vs. Generating Function .......... 23 2.2.3 Symplectic Maps as Lagrangian Submanifolds ......... 25 2.2.4 Functions as Lagrangian Submanifolds ............. 27 2.2.5 Existence of Infinitely Many Generating Functions ....... 31 2.3 Generating Functions from the Computational Point of View ..... 35 2.4 Computation of Symplectic Maps from Generating Functions ..... 40 3 The Symplectic Approximation Process 42 3.1 Symplectification of Taylor Maps and Symplectic Tracking by the Gen- erating Function Method ......................... 43 3.2 Transformation Properties of Generating Functions .......... 46 3.3 Equivalence Classes of Generating Functions .............. 50 3.3.1 Application to the Conventional Generating Function Types . 59 3.4 Equivalence of Symplectification with and without Linear Part . . . . 62 3.4.1 Equivalence in the Case of Symplectic Maps Conjugated by Linear Symplectic Maps ..................... 68 viii 4 Exam .1 15 3.5 Implementation .............................. 4 Examples 4.1 Maps Generated from Random Hamiltonians .............. 4.2 An Anharmonic Oscillator ........................ 4.3 An Exactly Symplectic Quadratic Map ................. 4.4 Muon Accelerators ............................ 4.4.1 A Neutrino Factory Lattice ................... 4.4.2 The FNAL Proton Driver .................... 4.5 The Large Hadron Collider (LHC) ................... 5 Optimal Symplectification 5.1 Formulation of the Problem ....................... 5.2 Hofer’s Metric and the Optimality Condition .............. 5.2.1 Hofer’s Metric ........................... 5.2.2 Connectedness of the Group of Hamiltonian Symplectic Maps 5.2.3 The Optimality Condition .................... 5.3 Link to the Extended Generating Function Theory .......... 5.3.1 The Fixed Point-Critical Point Relationship . . . ...... 5.3.2 The Generalized Hamilton-Jacobi Equation and Applications . 5.3.3 Siburg’s Theorem ........................ 5.3.4 Proof of the Main Theorem ................... 5.4 The Best Generating Function Type .................. 5.5 Some Further Remarks .......................... 6 Summary of Part I H Accurate Simulation of Fringe Field Effects 7 Introduction and Theoretical Background 8 Generic Effects 8.1 Aperture Dependent Effects on Linear Times and Chromaticities . . . 8.2 The Sharp Cutoff Limit ......................... 8.3 Shape Dependent Effects ......................... 8.4 Nonlinear Effects Due to Pseudo-Multipoles .............. 8.5 Symplectification ............................. ix 69 73 74 81 91 108 109 112 114 115 117 121 136 143 146 147 153 154 156 165 176 178 ,. .- ' I ‘- J :1 ‘ 1 I i , . . I . 9“ A], L I .;§ E . ‘: 115mm: B13110 8.6 Summary ................................. Multipole Decomposition of Magnets Represented by Wire-Currents 183 9.1 Biot-Savart Law and Field Computation ................ 9.2 Multipole Extraction Algorithms .................... 9.2.1 The Direct Method ........................ 9.2.2 Multipole Extraction by Analytical Fourier Transform ..... 9.3 Enforcing Maxwell’s Equations ..................... 9.3.1 Local Maxwellification ...................... 9.3.2 Global Maxwellification ..................... 9.4 Structure of B, and Bd, for Non-Maxwellian Fields ........... 9.5 Examples of Exact Multipole Decompositions ............. 9.6 Computation of Fringe Field Maps ................... 10 Fringe Field Effects on the Nonlinear Dynamics of the LHC 10.1 Methods of Analysis ........................... 10.2 Cases Studied ............................... 10.3 Results and Discussion .......................... 10.4 Map Transformations Under Orientation Flips ............. 11 Summary of Part II BIBLIOGRAPHY 184 190 190 192 200 216 217 219 221 246 253 256 List of Tables 3.1 Number of iterations needed for convergence in solving (3.175) for sev- eral examples, using the generator type associated with S = 0. The orbits of the particles of small, medium, and large amplitude, respec- tively, have been followed for 100 turns, and the number of iterations needed is enclosed in the intervals appearing in the table ........ 71 3.2 Number of iterations needed for convergence in solving (3.175) for sev- eral examples, using the conventional generator type F1. The orbits of the particles of small, medium, and large amplitude, respectively, have been followed for 100 turns, and the number of iterations needed is enclosed in the intervals appearing in the table. In the table, n.c. stands for no convergence in 100 iterations, and “not stable” means that the respective particle is predicted by the algorithm to be unstable. 71 8.1 Enge coefficients of the default fringe field falloff used in COSY INFINITY . for dipoles, quadrupoles, and sextupoles, respectively. ......... 155 8.2 Enge function with only two parameters, computed by slightly modify- ing the second default Enge coefficient (a2), such that the correspond- ing Enge function has the same integral as the default six parameter one. .................................... 170 8.3 Enge coefficients fitted for the LHC HGQ lead and return ends, respec- tively. ................................... 171 8.4 A few amplitude dependent tune shifts for the 25 mm aperture case. All 6 studied fringe field falloff shapes are included. .......... 172 8.5 A few amplitude dependent tune shifts for the 50 mm aperture case. All 6 studied fringe field falloff shapes are included. .......... 172 8.6 A few amplitude dependent tune shifts for the 75 mm aperture case. All 6 studied fringe field falloff shapes are included. .......... 172 8.7 A few amplitude dependent tune shifts for the 100 mm aperture case. All 6 studied fringe field falloff shapes are included. .......... 173 9.1 Results of transforming the analytic derivatives of the Biot-Savart law, computed with Mathematica, to Fortran code .............. 185 9.2 Computation time of the Taylor expansion of the y-component of the magnetic field in DA at various orders. ................. 186 xi ¥ 9.3 9.4 9.5 9.6 Taylor expansion of the magnetic field components 8,, and By. The columns represent the expansion coefficients, the order in the expansion and the exponents of (:r, s, y), respectively. ............... Comparison of the 3 component of the curl, up to order 3, computed by two different implementations of the Biot-Savart law ((9.11) and (9.6), respectively). ............................ The 3 component of the curl, up to order 12, after Maxwellification of the field in Table 9.3 ............................ Opening aberrations, (Ilan), for the return and lead ends’ exit focusing maps. ................................... 10.1 The various LHC realization cases studied. ............... xii 189 190 198 222 03 ‘n List of Figures 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 1000 turn tracking of a random two dimensional symplectic map with the 19th order Taylor map (considered to be essentially the exact result). 75 1000 turn tracking of the random two dimensional symplectic map obtained from the 19th order Taylor map, by symplectifying it with the generator type associated with the S = 0 symmetric matrix. . . 1000 turn tracking of the random two dimensional symplectic map with the 3, 7, and 11th order Taylor maps, and S = 0 symplectified map, respectively. ................................ 1000 turn tracking of the random two dimensional symplectic map with the symplectified maps utilizing the F1, F2, F3, F4 generating function types for the full 11th order Taylor map, and F2 and F3 for the nonlinear part of the 11th order Taylor map respectively ...... 1000 turn tracking of another random two dimensional symplectic map with the 19th order Taylor map (considered to be essentially the exact result) .................................... 1000 turn tracking of the second random two dimensional symplec- tic map with the 3, 7, 11, and 19th order S = 0 symplectified maps respectively. ................................ 1000 turn tracking of the second random two dimensional symplec- tic map with the symplectified maps using the conventional generator types F1, F2, F3, and F4 respectively. .................. 1000 turn tracking of a anharmonic oscillator with the 19th (considered to be essentially the exact result) and 7th order Taylor maps ...... 1000 turn tracking of an anharmonic oscillator with the 3, 7, and 11th order S = 0 symplectified maps respectively. .............. 1000 turn tracking of an anharmonic oscillator with the symplectified maps from the 7th order Taylor map, using the conventional generator types F1, F2, F3, and F4 respectively. .................. 1000 turn tracking of an anharmonic oscillator, with two random gen- erating function types associated with symmetric matrices with entries in [-1,1] and [—10,10] respectively .................... 1000 turn tracking of an anharmonic oscillator, with the generating function type associated with the symmetric matrix Sb, which was fitted to give the best results. ...................... 76 77 78 79 79 80 82 82 83 84 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 2000 turn tracking of an anharmonic oscillator, with the generating function type associated with the symmetric matrix S = 0, using the 11th order 1/2 time Taylor map ...................... Corrections introduced by the symplectification to particles of small amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = 0 to be on a invariant curve (a stable particle). The figure shows the correction with respect to the 7th order Taylor map of the symplectifications using the convential (the magnification factor is 103) as well as the S = 0 (the magnification factor is 106) generator types. .............. Corrections introduced by the symplectification to particles of medium amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = 0 to be on a invariant curve (a stable particle). The figure shows the correction with respect to the 7th order Taylor map of the symplectifications using the convential (the magnification factor is 1) as well as the S = O (the magnification factor is 102) generator types. .............. Corrections introduced by the symplectification to particles of large amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = O to be on a invariant curve (a stable particle). The figure shows the correction with respect to the 7th order Taylor map of the symplectifications using the convential (the magnification factor is 1) as well as the S r: (the magnification factor is 10) generator types .............. Global error propagation in the case of nonsymplectic versus symplec- tic tracking of a particle for 1000 turns of the anharmonic oscillator’s time one map. For the tracking, the order 7 Taylor map was used for nonsymplectic tracking, and generator of type associated with S = 0 for the symplectic tracking. After a short transient period, the non- symplectic method gives iarge errors, while the symplectic method’s error stays small. ............................. The global error propagation in a log-log scale of the case shown in Fig- ure 4.17. After a short transient period, the nonsymplectic method’s SIOpe is two, and the symplectic method’s slope is one. The dotted lines have exact slopes one and two, respectively ............. 1000 turn tracking of a quadratic symplectic map with the 3rd order Taylor map (the exact result), and the 3rd order S = 0 symplectified map (which also gives the exact result) .................. Symplectic tracking with the four conventional generator types (F1 through F4), for the exactly symplectic quadratic map, using fixed point iterations to solve the implicit equations .............. Symplectic tracking with the four conventional generator types (F1 through F4), for the exactly symplectic quadratic map, using Newton’s method to solve the implicit equations. ................. xiv 86 88 89 90 92 92 93 94 95 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 5.1 8.1 8.2 8.3 8.4 1000 turn tracking of the case 1 of a lattice of the proposed Neutrino Factory with the 8th order Taylor map, and the corresponding S = 0 symplectified map. ............................ 1000 turn tracking of the case 2 of a lattice of the proposed Neutrino Factory with the 8th order Taylor map, and the corresponding S = 0 symplectified map. ............................ 1000 turn symplectic tracking of the case 2 of a lattice of the pro- posed Neutrino Factory with the conventional generating functions (F1 through F4). ............................... 1000 turn tracking of the case 3 of a lattice of the proposed Neutrino Factory with the 8th order Taylor map, and the corresponding S = 0 symplectified map. ............................ 1000 turn tracking of the case 4 of a lattice of the proposed Neutrino Factory with the 8th order Taylor map, and the corresponding S = 0 symplectified map. ............................ 1000 turn tracking of the FNAL Proton Driver with the 15th order Taylor map, and the corresponding S = 0 symplectified map ...... 1000 turn (:r-a) tracking of the FNAL Proton Driver with the element- by-element numerical integration. .................... 105 turn tracking of the LHC with the 8th order Taylor map, and the corresponding various symplectified maps, without fringe fields taken into account. ...................... - ......... 105 turn tracking of the LHC with the 8th order Taylor map, and the corresponding various symplectified maps, with detailed fringe fields taken into account. ............................ 105 turn tracking of the LHC with the F2 and F3 symplectified nonlin- ear parts of the 8th order Taylor map, without fringe fields taken into account. .................................. 105 turn tracking of the LHC with the F2 and F3 symplectified nonlin‘ ear parts of the 8th order Taylor map, with detailed fringe fields taken into account. ............................... Plot of Hofer’s distance between the exact and the symplectified maps over the region enclosing the dynamic aperture of the anharmonic os- cillator studied in section 4.2. ...................... Center tunes as a function of aperture. The fringe field shape is given by the default Enge coefficients. ..................... Traces of the :r — a and y — b submatrices versus aperture. The fringe field shape is given by the default Enge coefficients. ......... First and second order a: Chromaticities as a function of aperture. The fringe field shape is given by the default Enge coefficients. ...... First and second order y Chromaticities as a function of aperture. The fringe field shape is given by the default Enge coefficients. ...... XV 97 97 98 101 102 156 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 9.1 9.2 9-3 Same as Fig. 8.1 (top), and Fig. 8.2 (bottom), but without the match- ing section, where most of the fringe field effects are concentrated. . . 160 Same as Fig. 8.3 (top), and Fig. 8.4 (bottom), but without the match- ing section, where most of the fringe field effects are concentrated. . . 161 Convergence to a constant of the second order (top) and blow-up of the fourth order (bottom) amplitude dependent tune shifts in the :1: tune. The fringe field shape is given by the default Enge coefficients. 166 Blow-up of the second (top) and fourth (bottom) order amplitude de- pendent tune shifts in the y tune. The fringe field shape is given by the default Enge coefficients. ...................... 167 Second order amplitude dependent tune shifts as a function of aperture in log-log scale. as tune shifts are on the top, and y tune shifts are on the bottom. The maximum lepe is around 1. ............. 168 Fourth order amplitude dependent tune shifts as a function of aperture in log-log scale. a: tune shifts are on the top, and y tune shifts are on the bottom. The maximum slope is around 3. ............. 169 Tracking pictures of on-energy particles launched along the x-axis with vanishing transversal momenta, magnet aperture of 75 mm, for all six types of fringe fields. From left to right and from top to bottom the following fringe field types are depicted: default dipole, default quadrupole, default sextupole, LHC HGQ lead end, two parameter Enge function, and GSI QD spectrograph. ............... 174 Resonance strengths of on-energy particles along the diagonal in action space, at a distance from the origin that corresponds to the approxi- mate dynamic aperture, magnet aperture of 75 mm, for all six types of fringe fields. From left to right and from top to bottom the follow- ing fringe field types are depicted: default dipole, default quadrupole, default sextupole, LHC HGQ lead end, two parameter Enge function, and GSI QD spectrograph ......................... 175 Tracking pictures of the Neutrino Factory, with the potential truncated in the pseudo- potential part at orders: 4 for (a), 6 for (b), and 8 for (c).177 Symplectic tracking picture (map and potential expansion at full order) of the Neutrino Factory for the same initial conditions as in Figure 8.13.177 Symplectic tracking pictures of the Proton Driver, for —4% momentum offset, (a) with full potential, and (b) with 6th order potential truncation.178 Tracking pictures of the Proton Driver without and with default fringe fields, and momentum offsets of 0 and :l:4%, respectively. ....... 181 Tracking of the Neutrino Factory without and with default fringe fields. 182 Lead End of the High Gradient Quadrupoles of the Large Hadron Col- lider ..................................... 206 Return End of the High Gradient Quadrupoles of the Large Hadron Collider. .................................. 206 Multipole strengths of the lead end as functions of 3. ......... 207 xvi 9.4 Normal multipole strengths of the return end as functions of s: b2(s), b5(s), b10(s), and b14(s). ......................... 208 9.5 The even order s-derivatives of the lead end’s b2(s). Shown are b2(2)(s), b2(4)(s), b2(6)(s), b2(8)(s), b2(1°)(s), and 1.203(3) .............. 209 9.6 The even order s-derivatives of the lead end’s b6(s). Shown are b6(2)(s), b5(4)(s), b5(6)(s), and b6(8)(s) ........................ 210 9.7 The even order s-derivatives of the lead end’s b10(s). Shown are b10(2)(s) and b10(4)(s) ................................. 210 9.8 The even order s-derivatives of the lead end’s a2(s). Shown are a2(2)(s), a2(4)(s), a2(6)(s), a2(8)(s), a2(1°)(s), and a2(12)(s) ............. 211 9.9 The even order s-derivatives of the lead end’s a6(s). Shown are a6(2)(s), a6(4)(s), a5(6)(8), and a6(8)(s). ...................... 212 9.10 The even order s-derivatives of the lead end’s (110(3). Shown are a10(2)(s) and a10(4)(s) ............................ 212 9.11 The even order s-derivatives of the return end’s b2(s). Shown are b2(2)(s), b2(4)(s), b2(6)(s), b2(8)(s), b2(10)(s), and b2(12)(s) ......... 213 9.12 The even order s-derivatives of the return end’s b6(s). Shown are b5(2)(s), b6(4)(s), b6(6)(s), and b6(8)(s). .................. 214 9.13 The even order s-derivatives of the return end’s b10(s). Shown are b10(2)(s) and b10(4)(s) ............................ 214 9.14 The lead end’s normal quadrupole strength, and its first and second s-derivatives, extracted from Maxwellian and non-Maxwellian fields, respectively, and superimposed on the same pictures. ......... 214 10.1 Tune footprints for the LHC case 1 .................... 224 10.2 Tune shifts for the LHC case 1. ..................... 224 10.3 Resonance strengths for the LHC case 1. ................ 225 10.4 Tune footprints for the LHC case 2 .................... 226 10.5 Tune shifts for the LHC case 2. ..................... 226 10.6 Resonance strengths for the LHC case 2. ................ 227 10.7 Tune footprints for the LHC case 3 .................... 228 10.8 Tune shifts for the LHC case 3. ..................... 228 10.9 Resonance strengths for the LHC case 3. ................ 229 10.10Tune footprints for the LHC case 4 .................... 230 xvii 10.11Tune shifts for the LHC case 4. ..................... 230 10.12Resonance strengths for the LHC case 4. ................ 231 10.13Tune footprints for the LHC case 5 .................... 232 10.14Tune shifts for the LHC case 5. ..................... 232 10.15Resonance strengths for the LHC case 5. ................ 233 10.16Resonance web for the LHC case 5, for d = 0. ............. 233 10.17Tune footprints for the LHC case 6 .................... 234 10.18Tune shifts for the LHC case 6. ..................... 234 10.19Resonance strengths for the LHC case 6. ................ 235 10.20Tune footprints for the LHC case 7 .................... 236 10.21Tune shifts for the LHC case 7. ..................... 236 10.22Resonance strengths for the LHC case 7. ................ 237 10.23Tune footprints for the LHC case 8 .................... 238 10.24Tune shifts for the LHC case 8. ..................... 238 10.25Resonance strengths for the LHC case 8. ................ 239 10.26Tune footprints for the LHC case 9 .................... 240 10.27Tune shifts for the LHC case 9. ..................... 240 10.28Resonance strengths for the LHC case 9. ................ 241 10.29Resonance web for the LHC case 9, for d = 0. ............. 241 10.30Tune footprints for the LHC case 10. .................. 242 10.31Tune shifts for the LHC case 10 ...................... 242 10.32Resonance strengths for the LHC case 10. ............... 243 10.33Resonance web for the LHC case 10, for d 2 ——2.535 and d = 0. . . . 244 10.34Dominating resonances for the LHC cases 9 and 10. .......... 245 xviii Part I Symplectic Approximation of Hamiltonian Flows 1" a . ‘ ' .. ' . . . . .. . DA ‘ \ V '. ' \y ‘ L ‘ \ | . ‘ ' ' v0 . a " i. ~ . ‘ . ‘ . . .. ‘A— A . . ,a 'v 0, ' ‘ ’ . I. "n A4. .v‘ I s, ‘ v ., ‘ u .. v. . 'lr . - . .h . .3. I -. I‘v.r 0" ,. l .i C \ Q ‘1 ‘A, .' ‘m. 'r ;,“v ._ .' e a a. . 0". W 5 ~. r. ,i v ‘a l s, V. . . .H , .fl !- v Chapter 1 Introduction A symplectic integration method is an integration method that preserves the sym- plectic structure (w : (15’ A d5, for more details see subsection 2.2.1) at every time step. It is well known that symplectic integration methods have favorable qualitative properties compared to non—symplectic ones, when used for long term integration. The good long term behavior has been explained by favorable global error prOpaga- tion (which is usually linear in the symplectic case, compared to generically quadratic in the non-symplectic case, and at least in certain cases stays bounded in the radial direction [1]), and the fact that the methods introduce only Hamiltonian perturba- tions of the original system; if the perturbations are small enough, according to the KAM theorem, most invariant tori, and hence most of the geometric structure, sur- vive [2]. Also, symplectic methods have very good energy conservation properties. Although it is known that in general the symplectic structure and the energy cannot be conserved simultaneously by a numerical method for a Hamiltonian system [3], the Hamiltonian is preserved by a symplectic integration scheme up to a function of the accuracy of the integrator, up to exponentially long times [4]. Sometimes (as a function of time step and initial condition) quasi-periodic and bounded energy er- rors are observed that seemingly last forever. There are various implementations of symplectic integrators in the fields of molecular dynamics [5], celestial mechanics [6], non—equilibrium statistical mechanics [7, 8], beam physics, etc. However, it is not clear geometrically what is the exact meaning of symplecticity. This is even more true for symplectic tracking with maps, as applied in the case of beam physics. On the one hand, the element by element symplectic integration usually is implemented in the thin lens (kick) approximation. While it is exactly symplectic, it is also slow, only accurate to first or second order in the time step, and not applicable in the case of general, nonseparable Hamiltonians, i.e. Hamiltonians that can be written as a sum of functions, which have flows that can be computed exactly. On the other hand, to assess the long-term stability of particles in a periodic accelerator structure in a reasonable amount of time, it is customary to compute an approximation of the one turn map, and then track with the map. Unfortunately, by approximating the originally symplectic map, for example by truncation of the Taylor series of the true map, its symplecticity is lost. Tracking large numbers of turns with truncated Taylor maps thus can potentially give inaccurate results. The best we can hope is that by recovering the exact symplecticity “artificially” from the truncated map, the long term tracking with the map will restore the properties of the original system, and will speed up considerably the estimation of the dynamic aperture. As we shall see, in the Differential Algebraic framework [9, 10] this can be done to very high orders. Therefore, tracking symplectically with high order maps is symplectic integration taken in its usual sense, but because of the use of integrators of very high order [10], it is usually more accurate in the time step and faster. It is also true that the speed is achieved at the expense of increasing the time step; here the time step is in fact one turn around the accelerator (using the arclength along the reference orbit as the independent variable). Sometimes it might be necessary to balance the length of the part of the system represented by a map with the required accuracy. This can be done by splitting the whole system into several pieces and representing each lump by a transfer map. Also, the map approach allows the important advantage of incorporating effects in the map that are otherwise very time consuming to compute, as, for example, fringe fields [11, 12, 13]. The main step in tracking symplectically with maps is the symplectification of the truncated, order n symplectic, Taylor maps. Several methods have been devel- oped to achieve the symplectification of maps. There are two main streams: one is based on factorization methods, and consist of Cremona symplectification [14], inte- grable polynomial factorization [15] and monomial factorization [16]; the other one is based on mixed variable generating function methods [17]. All methods provide valid symplectification schemes. However, the symplectified map depends on the spe- cific method used. It was realized that the particular schemes applied often make considerable differences in the final results. This realization triggered the studies of optimal symplectification. For details concerning optimal Cremona symplectification see [14].- In part I of this dissertation we extend the method of generating function symplectification to an exceedingly large class of generators, study the optimality of generating function symplectification expressed in terms of Hofer’s metric [18], and solve it by choosing the type of generating function that comes closest to satisfy the optimality condition in general. The first mention of the possibility of symplectic integration using generating func- tions dates back to 1956 [19]. Later it was rediscovered by others; see for example [20, 21]. Specifically, in beam physics, symplectic tracking with maps based on gener- ating functions was proposed in [17 , 16, 22]. In particular, it has been shown that in the Differential Algebraic framework it is straightforward to compute the order n + 1 truncation of the generating function from the order n truncation of the one turn map to any order n [17, 10]. Symplectic tracking to order three was first implemented in r um. I .u. . ,1- the code MARYLIE [23], and to arbitrary order it was first implemented in COSY IN- FINITY [24]. The possibility to estimate the SSC dynamic aperture with generating functions-based symplectic tracking with one turn maps has been considered in [25]. Another approach to generating functions and maps is based on fitted maps [26]. All these methods use only the conventional F1, ..., F4 (in Goldstein’s notation) types of generating functions [27]. Recently, a symplectic integration scheme has been developed that is based on generating functions, and it has been shown that actually there are infinitely many generating functions associated to a symplectic map [28, 29]. The methods of [28] are based on [30], which is basically a linear algebra problem, and its local generalizations to the nonlinear case. On the other hand, the rigorous mathematical foundation on manifolds of the classical global generating function theory has been laid in the 19708 [31, 32, 33, 34]. We combine the two, and give the general theory of generating functions of canonical transformations, with an eye on usefulness for computation in the Differential Algebraic framework [10] used in cosv INFINITY [24]. To be able to say which generating function is the best one first requires a char- acterization of the various types. This provides the motivation to develop the general theory of generating functions in chapter 2. First, it is shown that locally there is an isomorphism between symplectic and gradient maps, and this leads to infinitely many generating function types for every symplectic map. Then, the global theory is developed, which is based on transformation of the problem into a problem in sym- plectic geometry. This approach gives insight into various problems of locality versus globality of the generating functions, and emphasizes the generality of the approach. We mention that following Weinstein’s work, the geometric approach to generating functions in the physics literature appeared at various degrees of completeness, as, for example, in [35, 36, 37, 38]. Chapter 3 contains results related to the symplectification of truncated symplectic Taylor maps. Specifically, it is shown that the general theory of generating functions can be used to produce exactly symplectic maps. In fact, infinitely many can be pro- duced, each map being associated to a different generator type. It is shown that not every generator type produces distinct symplectified maps, and a certain subset of generators can be reduced to equivalence classes. Two types of generating functions are said to be equivalent if they produce exactly the same symplectified map, when applied to a given order n symplectic Taylor map. In this endeavor some transforma- tion properties of the generating functions are derived, which are interesting also in their own right. Also, there is a brief presentation of how the conventional generators fit into this framework. Sometimes it is preferred to factor out the linear part of the map to be symplec— tified, and apply the symplectification procedure to the nonlinear part only (in fact this part will have identity as linear part). It is proved that there is nothing to be gained by this approach if the appropriate types of generating functions are utilized. The implications of linear symplectic variable changes for the outcome of the sym- plectification process are also analyzed. The last section of the chapter gives some details about the implementation of the method in the code COSY INFINITY. Chapter 4 is devoted to a variety of examples. The tracking pictures obtained for standard test cases and accelerator lattices of practical interest are studied, and sev- eral generator symplectifications are compared. For a few cases, the local corrections introduced by the symplectification process to the Taylor maps are shown to exhibit unusual patterns. The examples of chapter 4 point out the necessity for optimal symplectification studies. In chapter 5 the problem is approached from the perspective of symplectic topology in general, and Hofer’s metric for compactly supported Hamiltonian sym- plectic maps in particular. In a very general way, the chapter gives a precise meaning to “the right way” to symplectify, and obtains a partial answer by singling out the generator type that is closest to satisfy the optimality condition in general. As by- products, a generalized Hamilton-Jacobi equation is found, which describes the time evolution of any generator type, and an interesting duality is developed between fixed points of symplectic maps and critical points of generating functions. Chapter 2 General Theory of Generating Functions of Canonical Transformations In the classical mechanics literature, traditionally only the 4 Goldstein type of gen- erating functions are well known. However, it is easy to show that, for example, the identity transformation cannot be generated by the type 1 and 4 generating functions. On the other hand, this set can be easily extended to 2'", or 4", generating functions. It can be showed that for any symplectic map, at least one generating function from this set exists locally [10]. The common factor for this set is that they all depend on mixed coordinates, more specifically on 77. initial coordinates and momenta, and 72. final coordinates and momenta. The first sign that in fact there are more generating functions dates back to Poincare, who used a different type of generating function, which not only is a mixed variable function in the sense discussed above, but also mixes (linearly) initial and final conditions in all the 27?. variables [39]. Later this generating function reappeared in [40] and [28], where also a unifying approach to the theory of generating functions has been presented, from which it resulted that there are infinitely many generating fllnCtions. However, while the approach of [28] gives important computational insight, the general mathematical foundation of the theory is contained in the series of papers [31, 32, 33, 40]. On the other hand, the general theory lacks exactly the computa- tional aspect. Our purpose is to give a rigorous account of the mathematical basis, and to cast the theory into a convenient computational tool within the framework of Differential Algebraic methods. We start with the local theory, since it already contains the main ideas, and it is easier to understand the underlying principles. The detailed account of the global theory follows in the next section, where light is shed on the somewhat obscure aspects of the local theory. 2.1 The Local Theory The theory is developed in Euclidean space, and all definitions and statements refer to this case. It will be generalized to arbitrary symplectic manifolds in the next section. First we introduce a few notations. Every map is regarded as a column vector. Let a=<$> an be a diffeomorphism of a subset of R4" onto its image, and let a-1 = ( Z; ) (2.2) be its inverse. Notice that a,- and Oz", 2' = 1, 2, are the first 2n and second 277. com- ponents of a and a“ respectively. This entails that a,- : U C R4" —> V,- C W", and analogously for (12'. It is worthwhile to note that there is a geometric significance to the use of R4". Both symplectic maps and functions under certain conditions can be given a geometric interpretation in the form of Lagrangian submanifolds of R4". (La- grangian submanifolds are 272 dimensional submanifolds of 472 dimensional symplectic v a. o -.. ' ' v b ‘h‘ manifolds on which the symplectic forms vanish identically.) Let A B Jac(a) = < C D ) (2.3) be the 4n x 471 Jacobian of a, split into 272 x 272 blocks. Let 7 _ J21: 0211 J4Tl _ ( 0211 —J2n ) a (2'4) where 0,, In J... — ( —I.. 0.. ), (2.5) and In is the unit matrix of appropriate dimension. A map a is called conformal symplectic if (Jac (0.))T .14., Jac (a) = tin, (2.6) where ,u is a non-zero real constant [41]. Also, we denote by I the identity map of appropriate dimension. A map M is called symplectic if its J acobian M satisfies the symplectic condition [42], that is MTJM = J. (2.7) Also, (2.7) can be written is several different forms [10]. We always assume that the symplectic maps are origin preserving. We call a map a gradient map if it has Symmetric J acobian N. It is well known that, at least over simply connected domains, gradient maps can be written as the gradient of a function (hence the name) [10], that is N = Jac (VF)T. (2.8) (VF is regarded as a row vector [43].) The function F is called the potential of the map. The best way to formulate the main result of this section is a theorem. 10 liar Dr E Theorem 1 Let M be a symplectic map. Then, for every point 2 there is a neigh- borhood of 2 such that M can be represented by functions F via the relation Mir-(muWWW where a is any conformal symplectic map such that det(C(M(z),z)~Mz+D(M(z),z)) 7&0. (2.10) Conversely, let F be a twice continuously diflerentiable function with gradient N, where N = Jac (VF)T. Then, the map M defined by M = (NC — A)‘1 (B — ND) (2.11) is symplectic. Definition 1 The function F is called the generating function of type a of M, and denoted by FOM. The theorem says that, once the generator type is fixed, locally there is a one-to- one correspondence between symplectic maps and scalar functions, which are unique up to an additive constant. The constant can be normalized to zero without loss of generality. Due to the fact that there exist uncountably many maps of the form (2.6), we can conclude that for each symplectic map one can construct infinitely many generating function types. A question that naturally arises is about the locality of the description of sym- plectic maps by generating functions, i.e. what is the size of the region where the generating functions are defined? A rigorous lower bound can be computed by com- bining the theory of this section with high-order Taylor model based verified methods. The results, using several examples of practical interest, suggest that at least certain 11 ¥ .,.o.. I" 1 _u. . 1.. Proof. ‘5 - . o- c»- .. p . . u to“ 'v H . P >- “-o u. v >- ' “- - u. . I, . ‘ .. \ a “ ‘4 no. la ' (l n, . n 7.1! - L. _,‘ _ . . i t ‘b .P ., ' A . ‘ . ll 4 t. x..' ‘\ carefully chosen generating function types’ domain of definition enclose the region of interest in simulations (for example the dynamic aperture) [44]. Proof. We notice that, by the implicit function theorem, the proof can be reduced to the linear case. In particular, the linearization of (2.9) at some point reads N = (AM+B) (CM+D)‘1. (2.12) Here all the entries in the equation are matrices. Therefore, by the implicit function theorem, if (2.9) is well defined at some point, i.e. det (CM + D) 75 0, then it also holds in a neighborhood of that point. Therefore, the proof is complete if we prove the following lemma. I Lemma 2 Let A, B,C, D E Rznfl", and 0.:(3 g). (2.13) Let M E R2nX2" be given. If A, B,C, D is chosen such that det (CM + D) ¢ 0, (2.14) and ifN E 112“?" is defined as N = (AM+B) (CM+D)", (2.15) which is equivalent to M = (NO — A)‘1 (B — ND), (2.16) then any two of the following statements imply the third one: 1) M is symplectic, i.e. MTJM = J, 2) N is symmetric, i.e. NT 2 N, 3) a is conformal symplectic, i.e. OTJ4nOz = p.74". 12 —— 1.. y' r i in t u . \ .,-u ~ ’- ”Li I ‘ r " ‘\ Papas PhMi ' o r 1 a - v , .. D-. u '. ‘ The proof is split into three part. First, it is shown that 1) and 2) implies 3), then that 2) and 3) implies 1), and finally that 1) and 3) implies 2). In the proof of the lemma we need the following proposition. Proposition 3 Let det (CM + D) 75 0 and N defined as in (2.15). Then det (NC — A) 75 0. (2.17) Proof. Taking determinants on both sides of (2.6) if follows that det (a) # 0. Thus, denote its inverse by _ A B Then, if we expand the relations Q ~ 0‘1 = (1'1 - a = I , we obtain AA+BC = AA+BC=LCB+DD=CB+DD=L (2.19) AB+BD = AB+BD=0,C/1+DC=C‘A+DC=O. (2.20) First we compute (0N + D) (CM + D) = [6 (AM + B) (CM + D)“1 + D] (CM + D) (2.21) = (7 (AM + B) + D (CM + D) (2.22) = (CA + Do) M + (GB + DD) (2.23) = I. (2.24) Taking determinants on both sides we obtain that det (CM + D) 74 0 => det (CW + D) ¢ 0. (2.25) Next consider the identity 11 0 I __N_ A B _ A—NC B—ND (226) c 1 0 CN + D C D — 0 I l 13 Taking determinants on both sides yet again, we obtain that det (CN + D) -det (a) = det (A — NC). (2.27) But det (a) # 0, hence det (CN + D) 76 0 => det (A — NC) 49 0. (2.28) Combining (2.25) and (2.28) we arrive at det (CM + D) 76 0 :> det (NC — A) ;£ 0, (2.29) and the proposition is proved. I We can proceed to prove the lemma. Proof. The first step is to prove that 1) + 2) => 3). We rewrite (2.12) to give N (CM + D): (AM + B). (2.30) Knowing that N is symmetric, transposition gives (MTCT + DT) N = (MTAT + BT) . (2.31) Combining (2.30) and (2.31) results that (MTCT + DT) (AM + B) (CM + D)‘1 = (MTAT + BT) (2.32) (MTCT + DT) (AM + B) = (MTAT + BT) (CM + D). Therefore we obtain that M" (CTA — ATC) M + (DTB — BTD) + (2.33) MT (CTB -— ATD) + (DTA -— BTC) M = 0. (2.34) The only way for this to hold for any symplectic M is by requiring that MT (CTA -— ATC) M + (DTB — BTD) = 0, (2.35) MT (CTB -— ATD) = 0, (2.36) (DTA — BTC) M = 0, (2.37) 14 for any symplectic M. This is true because the symplectic condition is a condition that involves quadratic relations among the entries of M, and these relations give as their results constants. From the first equation we conclude that it holds if and only if CTA-A'TC 2 DJ, (2.38) DTB—BTD = -pJ, (2.39) for any real number a different from zero, and the second and third equations hold simultaneously if and only if DTA — BTC = 0. (2.40) These relations can be cast into a convenient matrix form, namely (33)T(gg)(gg)-.(3 3,), 7..., where ,u E Rx. Hence, we can conclude that for any symmetric N and symplectic M there can be found nonsingular matrices A, B, C, and D such that (2.6) holds. Therefore, a is a conformal symplectic map. The second step is to show that 2) + 3) => 1). From (2.15) we can deduce that NT = (CM + D)’T (AM + B)T. (2.42) Using the assumption that N T = N, we get that (CM + D)‘T (AM + B)T = (AM + B) (CM + D)". (2.43) To remove the inverses, the above equation can be rewritten as (MTCT + DT) (AM + B) = (MTAT + 37’) (CM + D). (2.44) 15 1) '. . r;" "V~ it 14.1} 1 n.“- " 1 v . ‘ V, . .44. U .. \ "I .1, . - U} o, . ".4 '1 1 v ‘ l A” I ‘ ‘3 ‘ ‘Mi 1‘ ' l l [v A\w*~., ‘A 1 a a ’- Performing the Operations and regrouping of terms gives MT (CTA — ATC) M + MT (CTB — ATD) + (2.45) (DTA —- B‘C) M + (DTB — BTD) = 0. (2.46) From the expansion of (2.6) it follows that ATC - CTA = a], BTD — DTB = —11J, (2.47) ATC — CTA = 0, BTC — DTC = 0. (2.48) This entails that (2.45) reduces to MTJM = J, (2.49) showing that M is symplectic. To complete the proof now we need to show that 1) + 3) => 2). First we notice that, according to the above lemma, (2.15) always can be solved for M to give M = (NC — A)‘1 (B — ND). (2.50) Therefore, MT 2 (B — ND)T (NC — A)"T, (2.51) M-1 = (B — ND)‘1 (NC — A). (2.52) Also, from the symplectic condition MTJM = J it follows that JMT = M ‘IJ. Inserting (2.51) and (2.52) in this equation, gives J (B — ND)T (NC — A)“T = (B —- ND)‘1 (NC — A) J, (253) which can be expressed as (B —— ND) J (BT — DTNT) = (NC -— A) J (CTNT — AT) . (2.54) 16 ' V r 'VV 117$- . \"l .1 V “I".’ I _ pa , ~ . n l 1‘ Y‘ 1 u'j' v ‘ 1 ‘- 5‘. n l. . . '4\ ' .g * .. ,1 'VH r. 1 ‘u.‘ 1 u v , . I 1., I . vu, ‘ u‘ _ o. I... . .1 5., I -F\ ' v' r‘ t 'l ‘- .' '0 ‘ a. v -.' , v M.» I .-4 J '0 1 “v. “\. . 1.. . 0 x I. “.~ I. “.1' - / Rearrangement of terms gives N (CJCT — DJDT) NT — N (CJAT - DJBT) — (2.55) (AJCT — BJDT) NT + (AJAT — BJBT) : 0. (2.56) Next, we need to manipulate (2.6), which is equivalent to Q‘TJ = Dar—1 .7. Transposition gives Ja = pa’Tj, where we used that JT 2 -—J and TT = —j. Also, from J“1 = —J and j“ = —j it finally follows that def = )1]. (2.57) Expanding this relation yields AJAT — BJBT = 0, CJCT — DJDT = 0, (2.58) AJCT — BJDT = a], CJAT — DJBT = —)11. (2.59) As the last step, inserting these in (2.55) results in NT : N. (2.60) This completes the proof. I Theorem 1 has a simple, intuitive interpretation. It provides a way to construct infinitely many generating function types for any given symplectic map. The various types are parametrized by the group of conformal symplectic maps. For the existence of a certain type of generator, det (CM + D) ¢ 0 must hold. Conversely, given any function and a conformal symplectic map, theorem 1 provides a method for generation of symplectic maps. 2.2 The Global Theory The local theory is sufficient in most situations, as the cases of interest to us are weakly nonlinear Taylor maps around fixed points. Therefore, the treatment of practically 17 . 1 - fl or 1.) 1 0 "JA- “ ~4. r ' I \ D' .‘..45~ hr“ 9. u...» an. .- . ‘v .o— v u .. . c u > O ‘ 5. OJ . ,A. Y “ 1 .. c - , 4 ~- _ ~ 2' 0 1 .'_ ‘ ‘ o. , V‘ v i a..._ ' ‘ . ill I n- , ., ' 11.. ’ . f ' " . .1, l‘ ..‘ ' u: | ~ . if\'\ . A 1’) relevant systems is inherently local. However, there are important aspects that the local theory does not answer. For example, how can the theory be extended to symplectic manifolds? Moreover, the definition of N in (2.15) seems to be taken from thin air. Is there a deeper reason behind this definition? Or, is there a more general definition that provides even more generator types? Finally, what can be said about the domain of definition of the generators? Can a global generating function be defined for any symplectic map? In this section we try to answer these questions. The local theory shows that around any point local generators can be found. The question is whether the various local generators can be glued smoothly to form a global generator. Apparently, this cannot be done in general by generators of type linear 0. However, the existence of nonlinear conformal symplectic maps is not a priori obvious. Moreover, even if a global generator exists for a given symplectic map, does the same type of generator exist globally for other symplectic maps? The solution of these questions requires the “geometrization” of the problem, that is, reformulation of the problem into a problem in symplectic geometry. We begin with a brief introduction to symplectic geometry, the language used for the mathematical basis of the theory. More details can be found in [45, 46]. 2.2.1 Fundamentals of Symplectic Geometry Symplectic geometry is the natural mathematical language of classical mechanics, specifically Hamiltonian dynamics, as any variational principle can be given a sym- plectic interpretation. To begin, some fundamental concepts are introduced. Let P be a smooth manifold, and w a differential 2—form defined on it. If (12 is closed and non- degenerate, it is called a symplectic form, and the pair (P, w) is called a symplectic manifold. A form is called closed if its exterior differential vanishes, did 2 0. Closed- ness is a geometric constraint, which is equivalent to the Jacobi identity. On the other 18 hand, non-degeneracy is an algebraic condition. It means that at each point p E P the skew-symmetric bilinear form cop : TpP x TOP —> R which acts on tangent vectors, is non-degenerate. In terms of the associated linear map 5),, (v) (u) 2 top (v,u), we have that, if 62,, (v) (u) = 0 for each v E TpP, then u = 0. Hence dip : TPP —-> T;P is an isomorphism. This means that, relative to some local coordinate system around each point, the matrix of (I), and equivalently the matrix of (D has nonzero determinant. This in turn implies that any symplectic manifold is necessarily even dimensional, due to the fact that the determinant of any odd dimensional skew-symmetric matrix vanishes. A fundamental category of symplectic manifolds are the cotangent bundles of configuration manifolds, which are the phase spaces of dynamical systems. These manifolds carry a symplectic structure that is a generalization of the canonical sym- plectic structure of R2". On the Euclidean space itself, by identifying R2" with T *IR", we have a special coordinate system in which the symplectic form takes the simple form wo = dq’A dp‘. The coordinates ((7,117) are called canonical. This coordinate sys- tem is the symplectic counterpart of the orthonormal coordinate system of Euclidean geometry. The matrix of coo is denoted by J and has the form J=(_01 (1)) (2.61) where each entry represents a n x n block matrix, I being the appropriate unit matrix. We also note that the standard symplectic form can be defined in a coordinate-free way by (.00 = —d)\, where /\ is called the canonical one-form, and takes the coordinate representation /\ = 13’ - dq‘. Darboux’s theorem states that on any symplectic manifold such a coordinate system can be found in a neighborhood of any point, hence any symplectic manifold is locally equivalent (symplectomorphic) to the Euclidean space with its standard symplectic structure too. We call a symplectic form translationally 19 invariant if its matrix has the same form at any point on the manifold. Now we turn to symplectic transformations between symplectic manifolds. We will use hereafter interchangeably the notions of symplectic transformations, sym- plectic maps, canonical transformations, symplectic diffeomorphisms and symplecto- morphisms. By definition, a diffeomorphism M : P1 —> P2 between two symplectic manifolds (P1, col) and (P2, (.02) of the same dimension is called a symplectomorphism if it preserves the symplectic forms, that is M*UJ2 2 col, (2.62) where “ denotes the pull-back, which is defined as (M*w).(v1.vz) = cum.) (T.M - v1. TZM - 122), (2.63) where z 6 P1 and vl, v2 6 TmPl. In this case P1 and P2 are said to be symplectomor— phic. In canonical coordinates this definition takes the following form for a symplectic map of the Euclidean space with its standard symplectic structure we (Jac(M))T J (Jac(M)) = J. (2.64) Here Jac denotes the Jacobian and T the matrix transpose. The above definition can be extended to include conformal symplectic maps by the following relation MIW2 = 7‘ (M) W11 (2-65) With r (M) E RX. To see the significance of r, take a scaling map defined by afoul = Wm, B 6 IR", and apply it to both sides of (2.65). By choosing a = 1/ m, we obtain (M065sz 2 sgn (r (M))w1. Hence, if r > 0, then M045 is symplectic. If r < 0, then M033 is called antisymplectic. Essentially, it means that Mot/9 is orientation reversing. Actually, this is strictly true only if n is odd, otherwise, for n even, the Cartesian product (Mort) XI is orientation reversing; I being the identity map. This 20 . “7'. .‘.-I u. a - '..», o “haul. . s . ... -. ' 1 w . 8 . 1 “L10. . . . . , I. - , . u s ‘1. u , . s H \ 6. I I, l w r. ma 1 V ‘1 ..'. i . ‘ J‘. . ‘r ~\ ‘ ~ ‘k. l ‘| "r. ‘ .r v...‘ . F "ii follows from the definition of the symplectic map. Specifically, any symplectic map of a manifold into itself preserves the symplectic form, so it also preserves the volume form w" = wA Aw (n times). This is a never vanishing 2n—form on a 2n dimensional manifold, so the vector space it forms is 1 dimensional. Direct calculation shows that it integrates to a constant times the Euclidean volume. It follows that any symplectic manifold is oriented by the volume form, symplectic maps preserve orientation, and if global coordinates are available the determinant of the Jacobian of any symplectic map is equal to 1 at any point. Now we turn our attention to Hamiltonian systems, as the single particle dynamics in accelerators of interest to us can be described to a very good level of approximation by a Hamiltonian dynamical system. First we establish a few notations. Obviously, the symplectic maps of a symplectic manifold form an infinite dimensional Lie group under composition, denoted by Symp(P, w) E Symp( P), if it is clear which symplectic form is considered. Also its Lie algebra of symplectic vector fields will be denoted by X (P). In the view of non-degeneracy of symplectic forms, there is a one—to—one correspondence between vector fields and 1—forms via X (P) —> (21 (P) : X —+ i (X)w, (2.66) where we used 2 for the interior product. A vector field X is called symplectic if i (X) to is closed, that is d (z (X )w) = 0. By the Poincaré lemma, on connected manifolds every closed l-form is locally exact, so i (X) w can be written locally as the differential of a function i (X )w = dH. In this case the vector field is called locally Hamilto- nian. If dH exists globally (for example if the manifold is simply connected), X is called Hamiltonian, and H the Hamiltonian function. Conversely, for any function H : P ——> R, the vector field X H : P ——> TP determined by the identity 2 (X H) w = dH iS called the Hamiltonian vector field associated to the Hamiltonian function H. A 21 v' ' ...O (1.-.... u v '1 . 4‘ - L‘s. .- . o. ILA“... . . v1 "0 O r H u .1 1. yr .. ... . .. . w 'v Hamiltonian dynamical system is the triple (P, to, H). Hamiltonian vector fields form a Lie subalgebra of the Lie algebra of symplectic vector fields. The map X H -> H is a homomorphism. However, if we restrict ourselves to compactly supported Hamiltoni- ans, then the map becomes an isomorphism, and it can be viewed as a normalization condition, by specifying the arbitrary constant in H. With this normalization different Hamiltonians generate different flows. This will be important later in the definition of the Hofer metric on the space of compactly supported Hamiltonian symplecto- morphisms. Specifically, compact support means that the Hamiltonian and hence the associated vector field vanishes outside a compact subset. Recall that vanishing Hamiltonians generate the identity map. Hence, the support of a symplectic map is defined as the closure of the set where it is different from identity. To define Hamiltonian symplectomorphisms we need first the notion of isotopy. The time-dependent vector field X H“ associated to the time-dependent Hamiltonian H t at every t, generates a smooth l-parameter group of diffeomorphisms 653% satisfying d t __ X it .0 _ 21—t¢Ht _ Ht 0 0H1 ’ (DH: — I' (267) 451;, is called the Hamiltonian flow associated to Ht, or the Hamiltonian isotopy. A symplectomorphism 45 E Symp (P) is called Hamiltonian if there exists a Hamiltonian isotopy rt, 6 Symp (P) from do = I to (151 2 ob. We denote the space of Hamiltonian symplectic maps by Ham (P,w), or simply Ham (P) . It turns out that Ham (P) is a normal subgroup of Symp (P), and its Lie algebra is the Lie algebra of Hamiltonian vector fields. On simply connected manifolds Ham (P) is the identity component of Symp (P), that is any symplectic isotopy is Hamiltonian [41]. The group of Hamilto- nian symplectomorphisms is path connected. Also, any path in the space of Hamilto- nian maps is Hamiltonian. If a symplectic map is generated by compactly supported Hamiltonians, the symplectic map is also compactly supported, which means that it 22 \ u \ n,- \ v .. . .1 - l- . u . . .. 1. i. . . r . v I .H '0 1 l. as .. “It. s .\.. ..\. 5. y)... L f . 1 l l .1 F ni- vle 1.2 i. .7. .3. . . f. is the identity map outside a compact subset. To prove that the flows of Hamiltonian systems are symplectic we need the fol- lowing two formulae [47]: the Lie derivative formula d t it gi¢tw = $t£xtw, (268) and Cartan’s magic formula Ex,w = 2(Xt)dw+d(i(Xt)w). (2.69) Now it is straightforward to see that if do = 0 and (2 (Xt) w) = dHt, then fiqfiflo = 0, that is constant in time and equal to its value at t = 0. Hence we obtain (15:0) = w, for any t. The argument works also backwards, implying that if the flow of a dynamical system is symplectic then it is generated by Hamiltonian dynamical systems. 2.2.2 Primitive Function vs. Generating Function In this subsection it is shown that the main ideas of mixed variable generating func- tions are already built in in the symplectic condition. Consider symplectic transfor- mations, M, of a symplectic manifold (T*X,w). Let us assume that the manifold is simply connected, or in other words is an exact symplectic manifold. Then, every closed form is also exact. We can write to = —d)\. The symplectic condition takes the form d (A — M*)\) = 0, (2.70) from which follows the existence of a function F, such that /\ — M‘A : dF. (2.71) The function F is called the primitive function of M. However, there is no one—to- One correspondence between symplectic maps and primitive functions. Actually, the 23 ¥¥ " '12'1 l...L- . I 1‘ "'v* ;| 7' 7“! p: H .. . r. . ,4. x q " s . ' 'U-r.-.Q.', 1 *d...“ .u 6 F . ,. ' l \ “Lu“. 1 1 ‘ .9 .1. l, . I I ‘ £- 18- r . ‘ .5. Yr 0 .'. l 1. A 'I . '1 Q 6 .. D ‘1‘“,- .. ‘ ‘4 P: r l I v. ‘ “‘JD" ’ 1 \- " ‘- ‘ ‘ 2'1. T" ('9. .N . symplectic map is determined up to a left action (composition with M from the left) of an actionmorphism, i.e. a symplectic map that preserves the 1-form A. This can be easily seen by replacing M with another symplectic map, N o M, where N an actionmorphism. Therefore, we obtain A — (N o M)* A : /\ — M“ (N*A) (2.72) = )1 — M“). (2.73) = dF. (2.74) Hence, M and N oM have the same primitive function. All the N s with this property arise as lifts of diffeomorphisms on the base manifold [47]. Therefore, the primitive function determines the symplectic maps up to cotangent lifts. This is a manifestation of the coordinate independence of the symplectic condition, specifically A. It also implies that for one—to—one correspondence between M and F, F cannot be defined on the phase space. If we think of (5,15) as independent canonical coordinates, and M (6.15) = ((3, P), it follows from (2.71) that fi-dJ—P-dézdF(cj§fi). (2.75) Now, if the equation 63 = Q (c7213) can be solved for 13' to give a function F1 (q", 63) = F (5,13 (6', Q)), with (q‘, C) as independent variables, we obtain 8F1 (#1 Q) _. = p‘ , -—-————_, = — P, 2.76 a Q ( ) and we recognize it as the F1 (Goldstein type 1) generating function. The symplectic maps with this property are called twist maps. Now it is apparent that, in order to uniquely determine the symplectic map, a function must employ mixed variables, and this follows from the symplectic condition 24 -r ,, 0 ..U .m- ' 1 o - tn. ..,_v ‘0. _. I 6.- V . l " ‘ .. . ... , 1'1 v' v . r . I “A!“ n. "I .' ' v . ‘ . LA-‘_.“". m. 0,. . o- ' bu "‘ 1., 1 e. (if '. ‘“ A.“ [it ‘1. x.“ , ,y. itself. The method used in this section could be used to derive other types of gen- erating functions. However, there are many As such that w = —d/\, as for example A = —cj' - dp’ or /\ = %(15'- dri— cf- dp‘), etc. Moreover, different /\s can be chosen on the source and target manifold, and guessing is necessary for the variable changes. Therefore, it is not clear how much freedom is possible for construction of new gener- ating functions, and in general it is not convenient to work in this setting. We choose to work in another setting, which has been introduced by Weinstein. 2.2.3 Symplectic Maps as Lagrangian Submanifolds Initially, we have symplectic maps of a manifold in one hand, and functions on an- other manifold on the other hand. The local representative of the symplectic maps are vector functions of an even number of components. Our generating function is a scalar function. A priori, the most general method to connect the two in a one-to-one manner is not clear. In the early ’80s Weinstein formulated a “symplectic creed” with the motto “in symplectic geometry everything is a Lagrangian submanifold” (see defi- nition below). Indeed, in symplectic geometry Lagrangian submanifolds are the most important objects beside the symplectic manifolds themselves. These Lagrangian submanifolds provide the most general link between symplectic maps and generating functions. Both symplectic maps, and functions under certain conditions, can be put in one-to—one correspondence with Lagrangian submanifolds of appropriate sym- plectic manifolds. Once this correspondence is established, instead of working at the level of symplectic maps and functions we can work with Lagrangian submanifolds. At this point, the link we are looking for will be given by the most general type of diffeomorphisms that map these Lagrangian submanifolds into each other, or in other words the diffeomorphisms of identification of the two Lagrangian submanifolds. We will consider only submanifolds of symplectic manifolds that are properly 25 W l‘ n J o. \ I ~lb4 ‘4 “' v. 4 \ - U 1.1 1 v . , ~ ' fl 1 U ....' ‘ ‘. ‘,‘ \v‘i . f ,m V o. . H1“ “V I 1“: 1. e, 1 -< 1.. 1 1 aff- " 0 1.- 1 l 1 . ". biz‘i '- 4.“ » ‘ embedded and which inherit their topology from the ambient manifold. Also, we will need to work with injections of graphs of maps. It can be proved that the graph of a smooth map f : P1 —-> P2, Ffzilflplipllpepliflplep'zlcplxpz (2-77) is a smooth submanifold of P1 x P2 of dimension dim (P1) [47]. Moreover, if f is a diffeomorphism, then the projections 7r,- : Ff —> 13,-, i = 1,2 are diffeomorphisms [47, 32]. Lagrangian submanifolds are defined in terms of tangent spaces on which the symplectic form vanishes. Definition 2 Let (P, (D) be a 2n dimensional symplectic manifold and let L be a submanifold of P. L is called a Lagrangian submanifold if, at each 1) E L, TpL is a Lagrangian subspace of TpP, i.e. wPITpL E 0 and dim TpL = %dim TpP. Equivalently, if i : L ¢—+ P is the inclusion map, then L is Lagrangian if and only if i*w = 0 and dimL = §dim P. First we prove that any symplectic map M : (P1, 6J1) —> (P2, cog), M‘wz : col, can be interpreted as a Lagrangian submanifold in the Cartesian product space P2 x P1 with the symplectic structure u(7r§w1 — 7r‘1‘w2), where 77.,- : P2 x P1 —> P,-, i = 1,2, are the canonical projections and p. E RX. The graph of the symplectic map is the 2n-dimensional submanifold of P2 >< P1 PM = {(A4 (Z),Z) [z 6 P1}. (2.78) Denote P = P2 x P1 and w = ,u (773.111 — r’fwg); then we have the following: Theorem 4 M is a symplectomorphism if and only if I‘M is a Lagrangian subman- ifold of (P, w). 26 i": 3.7 L1". « i I)" . '9‘; ' ,4 ‘21 r,'1 _ f“, -. ‘ Proof n N *v , at .. . | 1. (h. n r .11 ‘ 'v. . l .1 .1 ' ’1 L. r..— 1 .. " .“\- .11 , I 1... ,. . lv 'I W 1.- 1. 1-. l " «.11; j, 1, A] Proof. PM is Lagrangian iff relative to the inclusion map i : I‘M H P, i‘w = 0. m = 11 (raga, — ”11.12) (2.79) = a ((772 o i)‘ w, — (77102)“ (dz) (2.80) = a (TM — M‘wg) (2.81) = u ((121 — M’wg) = O. (2.82) Hence, because ,u 7i 0, it follows that M’wg = to]. I Arbitrary Lagrangian submanifolds of (P,w) are called canonical relations, and can be considered as generalizations of symplectic maps. For practical applications we will be interested in the case P1 = P2 = R2", and to] = (122 2 too being the standard symplectic structure of R2". Thus, in this case, any symplectic map M : R2" —> R2" is a Lagrangian submanifold in (R4",uj), with symplectic structure to that has the matrix ” _ J2n 02n )1] — a ( 02” —J2n ) . (2.83) One particular Lagrangian submanifold of this kind that will be useful later is the diagonal, which by definition is the graph of the identity map A = {(2,2) [2 6 P1}. (2.84) 2.2.4 Functions as Lagrangian Submanifolds Next, we turn our attention to the one—to—one correspondence that can be set up between closed one-forms and Lagrangian submanifolds of cotangent bundles that project diffeomorphically onto the base manifold. Consider a smooth manifold X and a l-form defined on it. Regard it as a map from X to T "X . Then the following holds: 27 Proposition 5 The canonical one-form /\ E Q] (T*X) is uniquely characterized by the property that 0“/\ = o, (2.85) for every one-form o : X —> T*X. Proof. The a on the left hand side is considered as a map, and on the right hand side as a one-form. Denote the coordinates of T*X by (73:15). Then a regarded as a map in these coordinates can be written as o = ((7, f((j)) for some functions f,. Recall that the canonical one-form has the expression )1 = 13' - dcj'. Thus, using the definition of the pull-back we obtain (03)),7 = /\a(.r)'dq~0 (2.86) T = (‘37) (1.156) (2..., = 05'- (2.88) where we used that 36(5) = (50 f(cj)) -d (60 (j) = f0?) ~d(j’= oq~, (5,17) being regarded as components of the identity map. I Now we demonstrate that when a is closed, its image 0 (X) (o regarded as a map) is a Lagrangian submanifold of T‘X with the standard symplectic structure. Note that the graph I‘d of the one-form o is defined as Fa={(o(w),w) [wEX,o(w) ETJX}. (2.89) Its image by the inclusion i : I“, —-> T*X is Lagrangian. Relative to the projection 77 : [‘0 —+ X, I}, is uniquely determined by the one—form o if and only if 77 is a diffeomorphism. This can be seen from the fact that i = 0077. Hence, I“, is Lagrangian iff i‘wo = 0. Thus we get (a 0 7r)’ too = (o o n)‘ (—d/\) = —77* (d (0%)) (2.90) = —7l’*d0 = 0, (2°91) 28 v- v‘ v— r V '.. . . v 1.“ u n‘ I ‘1... .. v .Q‘ 1 , > . . . , 1'.- . ... - o. \ '- 3. ., 0Q . I u. , 1 u u, ,. 1 11 . v . a . ’1. 4... ‘ v a '5.’ ‘ N. H o r. u. , u. n b ‘I I .‘ where we used we 2 —d)\, the fact that the pull-back commutes with the exterior differential and the result of the above proposition. Therefore, there is a one-to-one correspondence between Lagrangian submanifolds of (TX, mug) of the form (2.89) which project diffeomorphically onto X, and closed one-forms 0 on X; here V E RX. In general, there are two ways to ensure that 7r is a diffeomorphism; if the em- bedding is 01 close enough (i.e. close enough in norm to the function values as well as the values of the first derivatives) to the canonical embedding of the zero section into the cotangent bundle, or a (as the differential of a function) is a diffeomorphism. . In the first case the projection mapping will be 01 close enough to identity to be a diffeomorphism. Indeed, it can be showed [45] that if 1 || df - I ll: -2-, (2.92) then f is a diffeomorphism. Hence, in this case 7r can be guaranteed to be a diffeo- morphism if a is Cl close enough to 0. For example, the zero section of T*X, defined by Z={(€,w) |§=0,w€X,£eT;,X}={0}> PdF ”ii i” (2.96) Pi 99‘; , X M being a symplectomorphism, the projections 7r, are diffeomorphisms. Moreover, it is assumed that 7r is a diffeomorphism, and there exist a diffeomorphism go,- : P,- —> X. Thus, in this case there exists a diffeomorphism a : I‘M —-> Pdp, for any pair of Lagrangian submanifolds of this form, satisfying the above conditions. The following theorem [48, 49] shows that 0 extends to a local symplectomorphism. It can be thought of as a generalization of the main theorem of [31]. Theorem 6 Let L,- be two Lagrangian submanifolds of the symplectic manifolds (13,-, wi), 2' = 1,2. Then any diffeomorphism a : L1 ——> L2 extends to a conformal symplecto- morphism B : U1 —> U2 of some neighborhoods U,- of L,- in P,, such that fiIL, E 01. Consider the following diagram: (P, w) B \ (T*X, z/wo) U4] IMF PM a > Pdp ngoiM/ \moiM 7r,/ \2' Z _.M__) M (Z) w L dF (10) Where 2: 6 P1 and w E X are arbitrary points of the respective manifolds. 32 ¥ w l A 4., .. 1‘ v \ ( . ‘, v. .4 We pointed out that the local classification, around a point, of symplectic struc- tures is completely solved by Darboux’s theorem. Here we have another concept of locality, namely local around a submanifold, of which one can think of as germs of the symplectic manifolds. Marle’s theorem is an example of such a tubular theorem, first proved to exist by Weinstein. It says that Lagrangian submanifolds do not have geometric invariants in order to distinguish each other. In general there are no global theorems similar to the tubular ones. This is true even in the R2" case, due to the existence of exotic symplectic structures proved by Gromov [50]. The importance of the above theorem consists of two main aspects: it gives the most general way to represent symplectic maps globally by scalar functions, and implies that once a symplectomorphism 6 exists for a symplectic map M and a function F, it is automatically valid for all nearby symplectic maps. It follows that the freedom for selecting generating function types is given by the set of conformal symplectic maps of the form [33.10 = w/u. A generating function of type a, which exists for a given M, exists for all nearby symplectic maps. In case the assumptions of the theorem are satisfied, any Lagrangian submanifold PM can be identified with any Lagrangian submanifold FdF; therefore, in principle the scalar function F can always be chosen in such a way that 1r is a diffeomorphism, guaranteeing global generating function types for any symplectic map. Clearly, once a is fixed, there is a one-to- one correspondence between symplectic maps and functions, hence they are called the generating functions of the symplectic maps. Interestingly enough, the diagram shows that, when the theorem’s conditions hold, to any pair (M, F) can be found an a (which is not unique) such that F becomes the generating function of type a of M. Also, generating functions can be defined on any simply connected manifold X diffeomorphic to P,, and as mentioned above, 6 cannot in general be extended to a global conformal symplectic map. 33 ...- .. ‘v . '1" ‘1. I IV 1.. ' 3’ Now we specialize these result to the case of Euclidean spaces. Notice that for our practical cases in beam dynamics M : R2" —> R2", and it is computationally conve- nient to define the generating functions also on R2". Hence P1 = P2 = X = R2", and col = wz 2 WO- As long as 7r is a diffeomorphism, the theorem applies and the exis- tence of the diffeomorphism a, which can be extended to a local symplectomorphism ,3 follows. Global canonical coordinates are available, in which the symplectic struc- tures w and mag are translationally invariant with matrices, pi and VJ respectively. Therefore, there is an a which can be used to identify any Lagrangian submanifold of (Rm, pi) of the form (2.78) with any Lagrangian submanifold of (Rm, VJ) of the form (2.89). According to the theorem, the most general form of 5 is a conformal symplectic map. Since a is the restriction of B to the Lagrangian submanifolds, we conclude that the most general form of the diffeomorphism a that links the symplectic maps to their generating functions is ,, 1 OZ LL10 '—"- ~01. (2.97) 1/ The coordinate expression of this equation is (2.6). In fact, this was expected from the local theory. Using elementary methods, it has been shown in section 2.1 that if (2.15) holds, the most general map that gives the generator type is a conformal symplectic matrix. The global theory just states that instead of matrices nonlinear maps can be used, and the local maps around each point can be glued together along germs of Lagrangian submanifolds, which entail the existence of global generating function types. In conclusion, we have the following fundamental results in the case of Euclidean space: there is a one-to—one correspondence between any small function (in the 02 sense) F E f (C°° (R2")) and symplectic map M : 1R2" —> R2", realized through a 34 . .u “Hui. ‘(v » - A“ "I. 5 .‘d I): ‘- diffeomorphism a : I‘M —> Four such that PdF = 0" (PM) » (2-98) 1 ofwo 2 Eco. (2.99) For a fixed a, the function F that satisfies the above equations is called the generating function of type a of the canonical transformation represented by M. The same type of generator exists for all symplectic maps close enough to each other (in the Cl topology). Obviously, from these equations it follows that to every symplectic map infinitely many generating functions can be constructed, due to the fact that the Lie group of diffeomorphisms of the form (2.99) contains infinitely many elements. This completes the global theory of generating functions. It is worthwhile to note that the theory is global in the sense that guarantees the existence of global generating functions for any symplectic map. However, it is local in the sense that it proves that one fixed type of generator cannot exist for all symplectic maps, not even locally around a point. 2.3 ’ Generating Functions from the Computational Point of View This section presents a computationally convenient method to obtain generating func- tions of given symplectic maps. First, it is necessary to rewrite (2.98—2.99) in a form that is convenient computationally. The vector function associated to the one-form dF by the standard Euclidean scalar product is the gradient, VF. We write N = (VF)T for the map, regarded as a column vector, represented by VF. “dN = 0” means that (W,- _aA/,- _ i 020,- 610i z',j = 1, ...,2n. (2.100) 35 C1 1'. \» 4- 1. These are the well-known necessary and sufficient conditions for the existence of a scalar potential [10]. The generating function is the potential of the closed one-form that determines the Lagrangian submanifold. Denote some canonical coordinates by z = (6,17), and denote the coordinates of the space where the generating function is defined by w. Introduce 2 and if) by 2 = M (2) and ”u“; = N (10). Then (2.98) can be expressed as (3)-“), (2.0.) ( N0”) ) = a( Mm ) (2.102) 211 \ z Splitting a into the first 2n and last 272 components, we obtain _ 01 a _ ( a2 ) (2.103) Similarly, for its inverse a“ we write _.1 (11 a _ ( a, > (2.104) From (2.102) it follows that w 2 ago( A; )(z), (2.105) N(w) = 01 o ( A; ) (2). (2.106) Combining the two equations we obtain that Noago('AI/i)(z)=alo(/¥)(z). (2.107) I forTnula, useful for the actual computation of the generating function: (VF)T= (alo( AI" )) 0 (a20( AI" D4. (2.108) 36 ReFo(ago('/;l )) (2.112) 37 There is a nice geometric interpretation of the global existence of the generating functions. As has been seen, any Lagrangian submanifold in unique correspondence with M can be sent diffeomorphically onto the Lagrangian submanifold determined by F. If, for some choice of a, this is achieved in such a way that PM is close enough to the zero section to project diffeomorphically onto the base, the generating function of type Oz exists globally. Recall that the group of diffeomorphisms is open in the set of smooth maps in the C'1 topology. Therefore, any smooth map close enough to the identity is a global diffeomorphism [41]. The reflection of this fact is that 02 (2, 2) has a global inverse. However, it might not be possible in practice to find an a satisfying this condition, especially for very nonlinear symplectic maps, and we need to consider local generating functions. If the projection diffeomorphisms are local, defined in a neighborhood of the origin, then we have local generating functions. In this case, one can think intuitively that fixing the type of generating function, as the nonlinearities of the symplectic map increase, the singularities move closer to the origin, limiting the domain of validity of the generating function. However, we always assume that the dynamics is taking place in a finite region of the phase space, so there is no loss of generality in assuming that the symplectic maps are compactly supported, and requiring only that the generating function to be defined in the region of interest. If the symplectic maps are too “big” for the generating functions to cover the region of interest, the problem can always be alleviated by taking roots of the symplectic map. Next, we are interested in the constraints imposed by oz. Equation (2.99) written in coordinates is 0;J4na# = j4n- (2.113) tl‘: By abuse of notation, for ,u/ 1/ we write )2 E RX. More explicitly (2.113) reads AT CT 02,, 12,, A B __ J2” 02,, f BT DT ) f 42.. 02.. C D ‘ “ 02. —J2.. ’ (2'1”) 38 which gives the constraints ATC—CTA = 11.12,, (2.115) BTD—DTB = —)i.12.,,, (2.116) ATD—CTB = 0. (2.117) Knowing that o is a diffeomorphisms, obviously a# is invertible. It is easy to see that det(J4n) = 1 and det (j...) = det(J2,,) seq—.12..) = 1, and hence (det(a#))2 = [14" ¢ 0. Then equivalently, (2.113) can be written as follows: 02;!“ = “Ana; (2.118) Transposition gives J4n0'# = #G;Tj4n, (2.119) where we used J}; 2 —J4,, and J; : —j4,,. Obviously, J47: = —J4n and L7} = —j4n, so it results that J4na#j4n = —,LLO';T, (2.120) J4na#j4na; = -,U., (2.121) a#.i4.,,a; =-. 0.14,, (2.122) which gives the equivalent set of constraints, but better suited for further analysis: AJgnAT—BJZnBT = 0, (2.123) CJgnCT—DJgnDT = 0, (2.124) DJQnBT—ngnAT = #12". (2.125) The constraints show that (A, B) and (C, D) must be symplectic pencils, and there is an additional condition that links the two pencils. An important observation is that in 39 the process of construction of the generating function from a symplectic map (2.123- 2125) have to be satisfied exactly, not only up to order n. In the Differential Algebraic framework, where we are working with Taylor expansions, it means that we need to construct as that are exact, at most order n, polynomial conformal symplectic maps. To our knowledge, their classification is not known. It follows that we are inherently constrained to consider linear maps a, which can be constructed and represented exactly on a computer. In this case a can be represented by a 4n x 4n constant matrix. If A is invertible, it follows that A‘IB is a symplectic matrix. The same argument holds for the case when C is invertible, resulting that C ‘ID is a symplectic matrix. Of course, even in the linear case, there are still infinitely many as to choose from. 2.4 Computation of Symplectic Maps from Gen- erating Functions In this section, the inverse problem is addressed. That is, given a generating function of type a, what is the symplectic map it generates? In other words, is there a “reversion” of (2.108) that gives M in terms of F and a? In section 2.1 this has been shown to be possible. However, it involves a somewhat awkward definition in terms of Jacobians. Here another method is presented, without involving the Jacobians. Introduce a transformation defined by W): (....( A; )) . (....( A; ))“, (2.2.) where Jac (a) 6 Cl (4n), and suppose that the transversality condition is satisfied. Therefore, TO (M) = N if and only if (N002 —al)o ( AZ" ) :0. (2.127) 40 Now suppose that M is given by another transformation T5 (IC) = M, where B :- (131,62)T, Jac (B) 6 Cl (4n), with the appropriate transversality condition satisfied. Inserting it in the above equation gives (Noag—a1)o (51°(§))°(52°(§))_1 =0. (2.128) I Because (B2 0 ()C,I)T) is invertible, this is equivalent to IC 61 O 1 I (No a2 — 01) o [C = 0, (2.129) B2 0 I 1, IC (NO (CY-2 O B) — (0'10 13)) O( I > = 0, (2.130) which in terms of T can be written as To, 0 T5 (K) = Tao/3 (,C), (2.131) for any IC. If we choose )3 = 0—1, it follows that Ta 0 T04 = T; = I, (2.132) that is Ta_. = (Ta)'1 . (2.133) This equation entails that whenever TO (M) = N is well defined, the inverse is automatically well defined, and gives M =Ta—1 (N). Explicitly, this means that M=(alo(§/))o(azo(¥))-l. (2.134) Applied to the situation where N is the gradient of the generating function and a is conformal symplectic, it gives a symplectic M. Thus, (2.134) is the counterpart of (2.9). Together, they provide a convenient computational method to pass from F to M and back. 41 n, .“‘ Chapter 3 The Symplectic Approximation Process Chapter 2 developed the general theory of generating functions and put the results into computationally convenient forms. It has been shown that, as long as the math- ematics can be performed exactly, the passing from symplectic maps to generating functions and back is easy to achieve. Moreover, this can be done utilizing any of the infinite set of generator types. Unfortunately, in practice this is not the case. Specifi- cally, the mathematical operations can be readily performed on a computer only with the truncation of the Taylor series of the maps at some order 12. Since there is a need to represent the accelerator systems by exactly symplectic maps for long term tracking purposes, procedures to recover the exact symplecticity of the truncated Tay- lor maps are needed. The symplectification can be achieved utilizing the generating function theory. This chapter studies various problems related to generating function symplectification. 42 3.1 Symplectification of Taylor Maps and Sym- plectic Tracking by the Generating Function Method Symplectic integrators became famous for their long—term properties. Their accuracy is not necessarily their best feature, and very high order symplectic integrators are often not efficient enough. The overall conclusion is that, for short term integration, the methods of choice should be integration methods with the smallest possible local error. For details see Sanz-Serna [1]. One turn around an accelerator can be con- sidered short term, and a good algorithm should give symplecticity close to machine precision over the region of interest. In the Differential Algebraic (DA) approach the integration of the reference particle also gives the truncated Taylor series of the one turn map, which is also close to machine precision order n symplecticity. The problems arise with the 108 — 109 iteration of the maps required for evaluation of the region of stable orbits, the so—called dynamic aperture. The errors potentially build up during large number of iterations, overshadowing completely the symplectic nature of the motion if sufficient time passes. Although, as it will be seen, recovering the symplectic nature of the motion is not a complete cure, there is reasonable hope that symplectic tracking captures the most important features of the original system over a sufficiently long time and sufficiently large region of phase space to be useful for applications. The symplectification process is the following. Using a given integration method, compute the truncated Taylor series of the map, M... It is assumed that Mn is order n symplectic. Fix an a that satisfies the constraints and the transversality condition. Use (2.108) to compute N", the truncation at order n of N. In COSY IN- FINITY all the necessary operations of truncated symplectic map generation, order n 43 composition and inversion of maps, and integration are readily available. Now, there is a fundamental difference between Mn and N". While in general M is loosing its symplectic property by truncation, because its elements are related by quadratic rela- tions, Nn still satisfies conditions (2.100) if N satisfies them, because they represent linear relations. Hence, a function which agrees with the true generating function’s Taylor expansion up to order n can be easily computed by integration of Nn along an arbitrary path. Therefore, if the true generating function is F, only E, can be computed. On the other hand, Fn is a valid generating function in its own right, and can be used to generate an exactly symplectic map. The Taylor expansion of the new symplectic map agrees up to order n with the original truncated symplectic map. Separating N" and a in linear and non-linear parts, we obtain N. = N,{:+N,{", (3.1) L+ N . = (30432.21) <32 Therefore, denoting by M, the exactly symplectic map generated by Nn (or F”), and E = M (z), (2.107) can be used to get (N: +N,,N) o (02" (2, z) + aév (E, 2)) = of“ (2, z) + a? (5, z). (3.3) The linear parts of can be written as matrices af(2,z) = 2+Bz, (3.4) 0%” (2, z) = C2 + Dz. (3.5) Isolating the linear part in 2 we obtain (N: o C — A) 2 = a1” (2, z) — N,” o (02 + D2 + a; (2, 2)) (3.6) —N,,L 0 (D2: + 39' (2, 2)) + B2. (3.7) 44 It already has been proved in section 2.1 that (NHL 0 C — A) is always invertible; thus a? (2, z) —N,,N 0 (C2 + D2 + a? (2,2)) :_ L __ '1 ‘_(N"OC A) O —N,[‘O(Dz+a.§v(2,z))+Bz (3.8) If the right hand side is contracting, it can be solved for 2 by iteration. Using as initial value 2,, = Mn (2:) = Mn (2) = 2", if n is sufficiently high, it is hoped that (3.8) is contracting, and it can be interpreted as a symplectic tracking scheme. The (k + 1)-th iterate of the map takes the form of, (2H1, 2k) — NnN 0(Czk+1+ 02" + 019’ (2H1, z“)) k 1_ L _ ’1 ' 2+ —(Nn 00 A) o _N#O(Dzk+a§\i(Zk+l’zk))+sz (3.9) Of course, at each (k + 1)-th iteration, it has to be solved for 2"“ as a function of 2“ by a fixed point iteration. With only linear maps a, (3.8) simplifies somewhat: 2:(NnLoC—A)“o[(B—NfoD)z—N,{Vo(cs+Dz)]. (3.10) Notice that (N: oC — A) is in fact a matrix. Denote N = Jac (Ni). It is actually the Jacobian of Nn at the origin. me (2.100) follows that N is symmetric. Interestingly, the first term appearing in (3.10) is the linear part of the symplectic map. Hence, assuming that the linear part of the solution can be computed always to machine precision, (3.10) takes the following form for the non-linear part of the solution 2"" 2" = - (NfOC—A)—10N,;N0(C2+Dz). (3.11) To this end, in principle the symplectification of maps and the symplectic tracking utilizing the generating function method is solved. We conclude the section with an interesting question. One of the reasons for tracking with maps is to speed up the simulation of accelerators. It seems that always implicit equations must be solved 45 . b" ’v . . 131 .. I..«¢ ‘U. , .— v- 3.? ’J.’ for tracking symplectically. If the fixed point iterations could be replaced by some explicit relations, probably the tracking would be much faster. This problem is equiv- alent to the question whether the generating functions must be of mixed variables? Unfortunately, in general the answer is yes. For any 0, denote Jac ((12) = (C D). If any generating function was non-mixed, it would mean that C = 0. But then it would follow that AB det (Jac ((1)) = det< 0 D ) = det (A) odet (D) (3.12) From the constraints (2123-2125) results that in this case (det (D))2 = (det (C))2 = 0, giving det (Jac ((1)) = 0, which contradicts the condition compatible with the constraints: (det (Jac (00)).2 = [14" ¢ 0. The same argument is valid for any block entry in Jac (0); hence A, B, C, D 75 0. In subsequent sections we will show that the complexity of the problem can be re- duced by considering equivalence classes of generating functions. However, the results of this section, taken together with subsection (3.3.1), is a recipe for transforming any code that uses the Goldstein generating function symplectic tracking to a form which uses fixed point iterations for solving the implicit equations, instead of the more tra- ditional Newton method. In section (3.5) it will be pointed out that the fixed point iteration has certain advantages over the Newton method. 3.2 Transformation Properties of Generating Func- tions By looking at how the generating functions transform under modifications of a and / or M, a set of rules is obtained, which are called transformation properties. These properties are based on the fact that if a is a conformal symplectic map such that (Jac (0))T J4" Jac (a) 2 him, (3.13) 46 :4 .. then for any 6 and 7 such that (Jac(6))TJ4nJac(fi) = .74., (3.14) (Jac()))Tj4nJac()) = .71.... (3.15) the map 5 o a o y is also a valid conformal symplectic map. Indeed, it follows from (3.14) and (3.15) and repeated application of the chain rule that [3 o a o 7 satisfies (3.13). Therefore, it gives another type of generating function. In invariant form, if a‘wo = aw, B‘wo = we, and 7*w = w, then (B o a o 1')‘ mm = 7* (0* (182%)) = 7“ (a*wo) = 7* (W) = W- (3-16) These rules are interesting in their own right in symplectic geometry, and some of them can be found in [29]. More importantly, the rules are used in the next sections for equivalence class reduction of the set of generating functions. We begin with studying what happens to the generating function FmM under the transformation (11 1—> Am, for some non-zero real )1. This affects only the conformality factor ,u of a, which becomes An. Slight rearrangement of (2.9) gives ((VFW)?’ o 32 — 0.1) o ( g“ ) = 0. (3.17) Then, we also have T M (VF(A£),M) oaz—A-al o( I )20, (3.18) which is equivalent to (v ( WM)”: .( A; ) = 0. .3...) 47 I -!V' (1. , ‘1...” A .p u . \ n' » - ‘ I ' . I) \ 1.1 7“- .p.. Comparing (3.17) with (3.19) we see that VFOM = v (1‘1 $080M) . (3.20) that is F(’\ao;l).M =/\‘Fa,M +6, (3.21) for some arbitrary constant c. In the same way, the transformation 02 H A02 has the following effect: ((VF(AaJ2),M)T0(A-ag)—a1)o( "I“ ) = 0, (3.22) «(varéwfsg _ ) .( ) (1-1. (V (F(Aac:2)‘MoAZ))Toog—al) o( ”I“ ) = 0, (3.24) (MA-1.(F(A.,2,,M.3)))T...-..).( ) Again, comparison of (3.17) and (3.132) gives = 0, (3.23) II S3 (3.25) v A“. F 8. AI = VFQ , 3.26 ( ( (A02)"M O )) [M ( ) that is F(’\Ol ),M =/\'Fa,MO)\—II+C. (3.27) 02 Also, from these two transformation properties it is easy to see the effect of the two transformations combined. Next, we study what happens if we change the symplectic map, for example, by 48 M +——> M o A, for some symplectic map A. From (2.9) we have ((VFmMoAf oag — 071) o < MIO’A ) = 0 (3.28) ((VFa,Mo.A)T 0 0'2 — 0'1) o ( if, ) = 0 (3.29) (T<> (0301:.) — (a. om) o ( é," ) = o. (3.30) where TA is defined by TA (2,2) = (2, A” (2)). Equation (3.30) can be written also as M ((VFaoTA,M)T O (02 0 TA) — (011 0 TA)) 0 ( I ) = O, (3.31) from where we conclude Fa,MoA = FaoTA,.M + C. (3.32) In the same manner, the left action of another symplectomorphism on the map, i.e. M +——> [C o M leads to ((VFQ,K0M)T 0 a2 —- 01) o ( [C DIM ) = 0. (3.33) Define TX; (2, z) 2 (1C (2) ,2). Then, FaJCoM = FQOTK,M + c. (3.34) We are also interested what happens when we change the coordinates in the gen— erating function, F l—> F 0 LI, by a diffeomorphism [. (here not necessarily a symplec- tomorphism); we have N3 ((V(Fo£))T 00/2 — (11) o< ) = 0 (3.35) (Jac (£))T- (VF)T o [I o 02 —- 071) o( A; ) = 0 (3.36) (VFT,.,,M)To(r032)— ((Jac(£))_T-a1))o( AI" ) = 0, (3.37) 49 '7'! \ ‘ u... L-- H ’l‘, HI» where we defined T; (2, z) = ((Jac (£))_T - 2, L (2)). Hence, V (FTCOQM o .6) = VFW”, (3.38) that is FR...“M = PM o 5‘1 + c. (3.39) Finally, if we replace M with M‘l, we arrive to ((VFQ,M-1)T o 02 -— a1) 0 ( M; ) = 0. (3.40) Applying M from the right we get ((VFQ,M_.)T 0 a2 — a1) 0( it ) = 0. (3.41) This is equivalent to say that, if in both 01 and 012 we interchange the first 2n variables with the second 272., we get back F M with this changed a. In terms of the Jacobian a#, or in case of linear as, this can be written as the transformation A B B A (CD)H(DC), (3.42) so we can conclude that + c. (3.43) 3.3 Equivalence Classes of Generating Functions We call two types of generating function equivalent if both types generate exactly the same symplectified map when applied to a truncated, order n symplectic, Taylor map. As we showed, the different types are parametrized by conformal symplectic maps. In this section we show that all the types generated by linear as, and which 50 exist at least locally for a given symplectic map, can be organized into equivalence classes characterized by symmetric matrices. Assume that for a given symplectic map M there exists a generating function of (VF)T= (alo( “21‘ D o (0.12% A14 ”A, (3.44) type a given by and Jac(a) = ( ’2. g ). (3.45) From (2.6) it readily follows that _1 _ JC'T —JAT (Jac(o)) — ( ——JDT JBT ). (3.46) First of all, it is straightforward to see that one can always change the conformality factor to p = 1 using the transformation rule (3.21), by choosing A = a“. From (2.134) and (3.46) it easily follows that we get the same symplectified map in both cases. Therefore, the conformality factor does not introduce any flexibility into the symplectification process. Hence we can always assume that ,u = 1, which is the most convenient value from the numerical implementation point of view. Denote the linear part of M by M. Then the generating function of the same type that generates the linear part M is given by (VF0)T=(a10( 1‘; ))O(020( A14))_1. (3.47) Subtraction of (3.47) from (3.44) gives (V(F—F0))T = [(0.1 — (VF0)T032) o( "I" H o (32% A; D7343) = (310(1‘1" ))o(ago(/\IA))—1=(VG)T, (3.49) 51 «$.33; u L... 4 un‘ . I Orfi 0' r . O '1'! .r ‘ . n . 4 ,7 c \ A u . - i . ’V . .i‘r“ G a 'u where we used the notations G = F — F0 and a, = a1 — (VF0)T 0 a2. Define B(1D,w) = ( “7’ (VF°)T(“’) ). (3.50) w Obviously, being a kick, i.e. changing only one component, 5 is a symplectic map for any function F0. Clearly, with 512 = (12 we have a = 5 o 0. Therefore, according to the transformation properties of the previous section, G is a valid generating function of type a. If we denote N = Jac (VF0)T 2 (AM + B) (CM + D)“, the Jacobian of a is given by _‘ _ A — NC B — ND Jac( )— ( C D ), (3.51) and its inverse by _' _1_ JCT —J (AT — CTN) (Jac (0)) — ( —JDT J (BT _ DTN) (352) Notice that N is actually the Hessian of a function, and hence symmetric, i.e. N T = N. Here we have to make an important observation. The symplectification procedure consists of starting with Mn and an a priori fixed a, and computing Fn using (2.9). Then (2.134) gives an exactly symplectic map, which we call the symplectified map. Unfortunately, on a computer (2.9) has to be represented by implicit equations and solved by fixed point iterations, but formally the Taylor expansion of the symplectified map (2.134) will be Mn up to order n. The point to be emphasized is that one needs an a priori fixed a that is exactly symplectic (not only up to order n) for the procedure to work. However, in general it is not easy to construct useful exactly symplectic polynomial maps of degree at most n. Even in the case that one constructs such a map, in general there is no reason to believe that (VF0)T as given by (3.47) will be a polynomial map of degree at most 71. Thus, in this case the exact symplecticity of it will be spoiled. Therefore, we are constrained to consider equivalence classes of 52 L.’ a. \'> .‘A‘ the types of generating functions associated with the subgroup of linear conformal symplectic maps. To this end, we can compare the two symplectified maps, that is the map obtained from E, and a, and the map obtained from Gn and Er. Notice that if a is linear, (VFO)T and hence a are also linear. Then for the Jacobians of the symplectified maps we obtain from (2.134) Jac(MF,,a) = [( JCT -JAT )( Jac(iFnfl )] . (3.53) [( _JDT JBT ) ( JaC(¥Fn)T )]—l (3.54) = (JCT Jac( (nTVF) —JAT)- (3.55) (— JDT Jac( V(F,,T) +JBT)—1, (3.55) .124ch,.) = [( JCT —)CTN) (Jac(V?)T‘N M (3.57) [ _JDT J(B _DTN) )( Jac(vaf—N H4958) = (.ICT Jac( (VF,,)T —JCTN—JAT+JCTN)- (3.59) (— JDT Jac( V(F,,)T +JDTN+JBT—JDTN)—l (3.50) = (.chT Jac( (VF,,)T —JAT)- (3.51) —JD ~Ja nT+JB _ . 3.52) ( T C(VF) T) 1 ( Since the maps are assumed to be origin preserving, we can conclude that MF 71,0 = MG (3.53) 7106. Thus we get the same symplectified map regardless of using E, of type a, or 0,, of type (1. So why is 0,, interesting? It is interesting because of the following property: 53 u. 2'! ,1 if we denote the Jacobian of 57 by A B (C D), (3...) from (3.51) we observe that AM + B : (A — NC) M + (B — ND) (3.55) 2 (AM + B) — N (CM + D) (3.66) = (AM + B) — (AM + B) (CM + D)‘1 (CM + D) (3.57) = 0. (3.53) Therefore we need to consider only the types that satisfy AM + B = O, in addition to the usual constraints imposed by (2.6). However, it is possible to further reduce the equivalence classes. We will use the transformation rule (3.39) with linear L. Denoting Jac (£1) = L and 67 2 Tc 0 53 we obtain -1T- -17“— Jac(€r)=((LL27A (LLI)DB)' We choose L = (CM + D)_1. After writing out explicitly the constraints contained (3.59) in (2.6), a straightforward calculation shows that (CM+D)‘1 = —M-1JAT, (3.70) (CM+D)T = — ITJA—l. (3.71) This entails that - —MTJ J Jac(“) “ ( —M-1JATC —M-1JATD ) ’ (3'72) and (Jac (Er (3.73) ))—1 _ JCTAMJ M _ —JDTA1WJ I 54 As mentioned above, from (3.39) we obtain that 659M 2 Farm 0 E. We drop the subscript, as there is no danger of confusion. Since .6 is linear, we also infer that G, = Fn o L, (3.74) and as a consequence (VGn)T 2 LT . (VFn)T o L, or (VF,)T = (L-1)T- (VC,)T 0 Cl. (3.75) We are now in position to compare the Jacobians of the two symplectified maps, and obtain T Jac(Mama) = [( (JCT —JAT ) ( Jac(YG") )] - (3.75) T -l [( —JDT —JMTAT ) ( Jac(VI70n) )] (3.77) = (JCT Jac (VG ,T) —JAT). (3.73) _ -1 (—JDT - Jac (VC,)T — JMTAT) , (3.79) where we used B 2 —/IM in the second line. Another calculation shows that ...1 T . . T . -1 Jac(MF,,5) ; JCTAJ( (MT) 1 M ) < (L ) Jac(VGn) L )] —JDTAMJ 1 )( (L-1)T.Jac(VG,,)T.L‘1 H4 (3.30) _ -l JDTAMJMTJA—1 Jac(VC,)T —M-1JAT) (3.32) = (JCTAJ( ()MT MTJA—T-Jac(VC,)T—MM-1JAT). (3.31) ( (JCT Jac( (,TVC) —JAT)- (3.33) ( _ —l JDT Jac( (VC,T) —JMTAT) , (3.34) 55 where we used M J M T = J. Hence, we obtained again that M0,,5 = Mpmd, (3.85) and after combining (3.63) and (3.85) we finally arrive at MF,.a = MF,,5- (3.86) Thus, the symplectified map obtained from a truncated generating function of type (linear) a agrees with the symplectified map obtained from type 51. Denote .. A B Notice that the property from the first step of the reduction, that is ~ ~ AM + B = —MTJM + J (3.33) = —J + J = 0 (3.39) is preserved, and in addition it has another very nice property, namely ~ ~ CM + D = —M-1JAT (CM + D) (3.90) = —M“JAT (AT)‘1JM = 1, (391) Where we used (3.70) in the second line. Therefore, every generating function type associated with linear maps, which ex- iStS at least locally for a given symplectic map, is equivalent for symplectification Purposes with another type associated with A B ( C D ) , (3.92) such that the following relations hold: AM+B = 0, (3.93) 56 Equations (3.93) and (3.94) have to be imposed in addition to the usual constraints derived from (2.6), that is AJAT — BJBT = 0, CJCT — DJDT = 0, DJBT — CJAT = 1. (3.95) These five conditions restrict very much the pool of independent generator types. From (3.93) and (3.94) we obtain B 2 —AM and D = I — CM respectively, which inserted in (3.95) gives A = —JM"1, (3.96) and re-inserted in (3.93) gives B 2 —AM : — (—JM"1) M = J. (3.97) The first condition in (3.95) is automatically satisfied if we impose (3.93) and (3.94). Inserting D = I — CM in the second relation of (3.95) we obtain CMJ — (CMJ)T = J. (3.93) We make the ansatz 1 CM 2 5(1 + JS). (3.99) This is always possible for some matrix S. Insertion of (3.99) in (3.98) gives that 5 must be symmetric, i.e. ST = S, (3.100) but can otherwise be arbitrary. Thus we obtained = — (1 + JS) M", (3.101) [0le H (I - JS)- (3.102) 57 Therefore, every generator type belongs to an equivalence class [S] associated with —J M "1 J ( g— (1 + .18) M-1 § (1 — JS) ) ’ (3103) and represented by the symmetric matrix S. Given an arbitrary type of generating function, how do we know which equivalence class it belongs to? We saw that C = (CM+D)"1C', D = (CM +D)-1D, (3.104) and similarly ~ 1 _1 ~ 1 C=§(I+JS)M ,D=§(I—JS). (3.105) We can express CM — D from the first two and second two relations respectively, obtaining (CM + D)‘1 (CM — D) = Js, (3.105) or equivalently s = —J (CM + D)"1 (CM — D). (3.107) To remind ourselves, equivalence means that generating functions from the same equivalence class will produce indistinguishable results if used to symplectify a given order n symplectic map. Thus we just proved the following theorem. Theorem 7 Every generating function type associated with a linear conformal sym- plectic map that exists at least locally for a given symplectic map belongs to an equiv- alence class represented by a symmetric matrix. An arbitrary type of generator, as- sociated with a linear a satisfying conditions (3.95) and (2.111), belongs to a class associated with (3.103), and characterized by the symmetric matrix given by (3.107). 58 3.3.1 Application to the Conventional Generating Function Types To exemplify the process, we show how the general theory contains the traditional, Goldstein type, generating functions. Because of (3.107 ), without loss of generality we can assume that the symplectic maps have identity as linear part. More explicitly, fixing the linear part does not affect the calculation of the matrices A, B, C, and D, but it does influence the calculation of S. However, for a different linear part, use of (3.107) will give the appropriate matrix S in each case. Here we arbitrarily fix the linear part to identity to exemplify the calculation process. In canonical coordinates (5,15), an origin preserving symplectic map acts as 5 Q' _, = _. , .1 M ( p ) ( P ) (3 08) The type F1 is the solution of the implicit relations T (7 _ 15' (V171) (Q ) — ( _13 ). (3.109) Clearly, in this case a can be chosen as a linear map, so in view of (2.107) we have (VF)T(C(%)+D(g))=A(%)+B(g). (3.110) Wechoose A = (3:31), B=(8:(I):), (3.111) (1:638) D=(é:g) (3.112) These matrices satisfy the constraints (2123-2125) and clearly substitution into (3.110) gives (3.109). We say that the F1 type of generating function is associated to 0,, 0,, 0,, I, 0, —I, 0,, 0n Jac (a1) = On On In On , (3.113) I, 0,, 0,, 0, 59 or that F1 is of type a1. Similarly, the Goldstein type 2 generating function F2, given by .....(g)-(,), is associated with 0,, 0,, 0,, In In 0,, 0,, 0n Jac ((12) = on on In 0n 0,, In 0,, 0n QHQ mum It can be easily checked from (3.107) that it belongs to the class represented by 0 I 3=(, 0). The conventional type three (F3 determined by L MAMA—7%) is associated with 0 0 —I 0 0 —I 0 0 Jac (a3) = 0 0 0 I , I 0 0 0 and belongs to the class 0 I $‘(1 0)’ differing only by a sign from the type F2 S3 = —Sg. Finally, the conventional type four (F4) is determined by ) = ( T) Q 7 60 W1 ”31 (VF4lT ( (3.115) mun (3.113) (3.119) (3.120) mun and is associated with Jac (a4) = (3.122) OONO NOOO COO ONOO It is well-known that the type F1 cannot be used. in the case when M = I. This naturally follows from the general theory, due to the fact that the transversality condition is not satisfied. That is, det (C - I + D) = det< f," 8" ) = 0. (3.123) Therefore, it does not belong to any equivalence class for symplectic maps having identity as linear part. The transversality condition is also violated in the case of the F4. On the other hand, in the case of the F2 the transversality condition is satisfied for the identity map, as expected det(C-I+D)- —det<0 " 3”)=1 . (3.124) An analogous result is found for F3 too. Thus, we recovered from the general theory the well known facts that F1 and F4 cannot, while F2 and F3 can be used to represent at least locally symplectic maps having identity as linear part. Also, we identified the equivalence classes which F2 and F3 belong to. The only difference if the symplectic maps do not have identity as linear part is that we obtain different symmetric matrices, and hence classes, which also can be computed using (3.107). 61 3 top L. . l I . g V Yllt I; . 3.4 Equivalence of Symplectification with and with— out Linear Part Symplectification can be performed on the nonlinear part only by first factoring out the linear part, or on the whole map. The next question that arises naturally is whether there is something to be gained if one first factors out the linear part of the symplectic map to be symplectified. This section answers the question in the negative. Moreover, combining results of this section, we show in subsection 3.4.1 that not even a linear symplectic change of variables can provide additional freedom in the symplectification process. We write the symplectic map to be symplectified as M = M + H, (3.125) where M is the linear part, and H the higher order terms. We can distinguish three symplectification procedures: symplectify M, directly, symplectify M L,” obtained from ML=I+M_10H, (3.126) or symplectify M Rm obtained from MR=I+HoM‘1. (3.127) In the latter two cases we first factored out the linear part from the left and right respectively. The relations among the maps are the following: M = MOML, (3.128) M = MRoM. (3.129) The question is whether these relations continue to hold for the symplectified versions of Mn, M L‘n, and M R.n- Suppose we symplectify the maps using a generator of type 62 a with Jac(a) : < 21, g ) . (3.130) The local existence conditions are det (CM + D) ¢ 0 (3.131) in the first case, and det (C + D) are 0 (3.132) in the second and third case. Being linear, we use M for the Jacobian of M too. Equations (3.131) and (3.132) are not compatible in general. Thus in general not every type of generating function exists in all three cases. The right question to ask is the following. Suppose one uses some type of generator to symplectify a given map, using the approach of one of the three cases. Then, are there other types of generators which produce the same symplectification for the other two cases? In other words, we would like to find the appropriate type of generators such that relations (3.128) and (3.129) hold for the symplectified maps. To this end, any generator for the first case is associated to one of the following: _JM-1 J Its inverse is given by 1.. 2 (Jae ((1)) " ( _%J(I+SJ) I ) (3.134) Denoting the generator by F5], Jac (VF[5])T =,, M5], and the symplectification of 63 Mn by M[3] we obtain Jac(/Mm) = [( 511110—51) M)(N}ST )] (3.135) [(-5J(1+SJ> I)(1‘)ST)] (3.136) = M-(§J-N[s,+1)- (3.137) (—§J-N[51+1)"l. (3.138) Now we turn our attention to the second case. Here the possible generators belong to one of the following classes: —J J Clearly its inverse is 1 _ —1 = §J(I‘S;]) I (J99 (13)) ( _%J(1+SJ) I . (3.140) Again, denoting the symplectified version of M L,” by M143] we obtain Jac(ML[5-.]) = -( §J(1—STJ) I ) ( 1‘13] H . (3.141) I —1 ( —)J(I+sJ) I ) ( NF] H (3.142) = Gun—SJ) -N[5.] +1) (3143) (—1J(1+sJ) -N- +1)—1 (3144) 2 [S] ’ ' T where we used the notation Jac (VF[S]) 2,, lel Next, we use the transformation 64 property (3.34) with IC : M ‘1. It follows that FBML = FfionM. (3.145) Also notice that B o T IC = a if and only if S = S. In this case NF] = M5]. (3.145) Comparing (3.137) and (3.143), and using (3.146) we can conclude that MW 2 M o MLF], (3.147) if and only if S = S. (3.143) This proves that the symplectified version of (3.128) is (3.147), and holds only if (3.148) is satisfied. We can proceed to the third case and follow the same route. To symplectify M R," we choose a generator type from the pool Jac(q) = ( 1 (I—J ~ 1 J ~ ) , (3.149) 5 + Js) ,(1 — Js) with inverse Jac , ‘1 = f] 0—3?) I . 3.150 ( (7)) —%J(I+SJ) I ( l 65 If we denote the symplectification of M 3,, by M ngl we get Jac (MR[§]) 1 ~ —1 —5J (I + SJ) «([5] + I) , (3.154) T using the notation Jac (VFISI) 2,, lel Now using the transformation rule (3.32) with A = M“1 we obtain F‘NMR = F70TA,JM- (3.155) A straightforward calculation gives that Jac(yoTA) = ( 1 (1:25,) 1 (13315,)1‘4 ), (3.156) 2 2 with inverse (Jac (T) o CZ“,2())_1 = ( %J (I — SJ). I \ . (3.157) —-;-M‘1J(I+SJ) M-1 ) Then, (3.153) can be expressed as Jac (MR[§]) = X . M-T, (3.153) where we introduced the notation 1 " 1 —1 ” —1 _1 X : -2—J (I — SJ) .VF,,T,,M + I . -—§M J (I + SJ) .VF,,T,,M + M . (3.159) 66 P\ [1+ But as one can see from (3.157) this is nothing else than the Jacobian of the sym- plectified map obtained from M, and generator of type (3.156). As shown in section 3.3 this generator type for M, belongs to the equivalence class represented by the symmetric matrix calculated using formula (3.107). A short calculation gives the result, S = MTSM. (3.150) Combining this result with (3.158) and (3.159) we can conclude that M[5] = M45] 0 M. (3.161) This proves that the symplectified version of (3.129) is (3.161), and holds only if (3.160) is satisfied. Therefore, the main result of this section can be formulated as the following theorem. Theorem 8 The symplectified version of (3.128), i.e. (3.147), holds if and only if (3.148) is satisfied, and the symplectified version of (3.129), i.e. (3.161), holds if and only if {3.160) is satisfied. The main point we learned is that from the optimal symplectification point of view there is no difference which way one proceeds. Once we obtained the best type of generator for one case, the best generators for the other cases automatically follow from (3.148) and (3.160). Therefore, there is nothing to be gained by factoring out linear parts and symplectifying the nonlinear parts only. Moreover, the first case (without factorization) is the most efficient when implemented numerically on a computer. 67 3.4.1 Equivalence in the Case of Symplectic Maps Conju- gated by Linear Symplectic Maps Combining the left and right factorizations in linear and nonlinear parts just discussed, we can address the special case of the linear symplectic change of variables. From (3.128) and (3.129) we can infer that MHzMoMLoM_1, (3.162) and from (3.147) and (3.161) that Map] = M 0 M4510 M“, (3.153) if S = MTSM. (3.154) The two maps are conjugated by a linear symplectic transformation. However, this case is very special, since both M R and M L are obtained from the same map M. We could relax the conditions, and ask if any two symplectic maps are conjugated by an arbitrary linear symplectic map, then the same holds true for their symplecified counterparts. That is, suppose that M and N are symplectic maps such that NleoMolC‘l, (3.155) for some linear symplectic IC. The possible types of generating functions are associated with —JKM--TK-1 J Jac (a) _ ( )0 + JS)KM‘1K‘1 §(1 — JS) ) (3'166) for N", and —JM-1 J Jae”) _ ( 3(1 + JS) M-1 )(1 — JS) ) (3'167) 68 for Mn. As before, we denoted M = M + H and Jac (IC) = K. The transformation rule to be used here is FaJV = F307,,M, (3.153) where Jac(TK) = ( I; If > . (3.169) Following now well established procedures, it is straightforward to Show that MS]: ICoM[§]oIC'1 (3.170) holds for the symplectified maps if S = KTSK. (3.171) Therefore, there is no additional freedom in the symplectification process if the sym- plectic map is first subjected to a linear symplectic variable change. 3.5 Implementation In this section, the implementation of the extended generating function symplectifi- cation method to COSY INFINITY is described. The method starts with M, given, and some arbitrary initial condition 2. Utilizing (2.9) with a given by (3.103), the truncated a-generating function F,“ is obtained. The arbitrary symmetric matrix S must be specified, fixing the type of generator utilized. All the necessary operations of map composition, map inversion, differentiation and integration are readily available in COSY. Then, notice that (2.9) can be expressed as 2—M-z=M-J-(VF,)T(C-(2—M-z)+2), (3.172) 69 l0 where we denoted 2 = M[5] (z), MlSl representing formally the symplectified map, and C = —;-(1 + JS)M-1. (3.173) To avoid as much as possible any problems with cancellation of digits, we denote w = 2 —- M - 2, (3.174) which leads to w = M - J - (V717,)T (C - w + 2). (3.175) This can be solved by a fixed point iteration for w, and gives the final result by 2 = w + M - 2. (3.176) The orbit of z is then computed by iteration of the procedure (in the next step we take 2 as the initial condition, etc). Writing (3.175) as w = f (w), we observe that to be able to solve (3.175) by a fixed point iteration, it is sufficient (but not necessary) for the right hand side to be contracting for a fixed 2, i.e. is guaranteed to succeed if If ('wz) - f (w1)l S q - lwz — wil, (3.177) for some q < 1. To have a good chance of contractivity over an extended region, the derivatives of the generating functions must be small. From our experience, in general the fixed point iteration converges in the region where the generating function is defined. We compiled a table with the number of iterations needed for convergence, for a few relevant examples that will be presented in more detail in the next chapter. The examples that we tracked are: two maps generated from random Hamiltonians, an 70 Using S = 0 Example Number of iterations Small ampl. Medium ampl. Large ampl. Random map 1 1 [4,10] :9, 29: Random map 2 1 [1,9] :3, 34 Anharmonic oscillator 1 [1,8] :1, 28 Quadratic map 1 1 1 LHC [4, 5] [7, 10] [8,68] Table 3.1: Number of iterations needed for convergence in solving (3.175) for several examples, using the generator type associated with S = 0. The orbits of the particles of small, medium, and large amplitude, respectively, have been followed for 100 turns, and the number of iterations needed is enclosed in the intervals appearing in the table. Using F1 Example Number of iterations Small ampl. Medium ampl. Large ampl. Random map 1 :1, 4] [6, 36: [7, 56] Random map 2 :1, 8| [3, 46: not stable Anharmonic oscillator 1 [1,8] [1,27] Quadratic map [4, 15] [13, no] [7, no] LHC [4,7] [7, 16] not stable Table 3.2: Number of iterations needed for convergence in solving (3.175) for several examples, using the conventional generator type F1. The orbits of the particles of small, medium, and large amplitude, respectively, have been followed for 100 turns, and the number of iterations needed is enclosed in the intervals appearing in the table. In the table, n.c. stands for no convergence in 100 iterations, and “not stable” means that the respective particle is predicted by the algorithm to be unstable. anharmonic oscillator, an exactly symplectic quadratic maps, and the Large Hadron Collider with fringe fields included. For more details see next chapter. The results are summed up in Table 3.1 (using the generator associated with S = 0) and Table 3.2 (using the conventional F1 type). Of course, (3.175) can be expressed as f (w) —— w = 0, and solved for w by New- ton’s method. We noticed that the results not only are sometimes dependent on the generating function type employed, but also on the numerics, that is the particular numerical method used to solve the implicit equations. Of course, if we start with an 71 exactly order n symplectic map, and the convergence to the solution of the implicit equations is achieved over the tracking region for both methods, then the resulting pictures are identical. However, in practice the fixed point iteration works in a more stable manner. It is faster than Newton’s method when many particles are tracked si- multaneously, and, in the vast majority of cases studied, its domain of convergence is larger. For maps of practical interest, Newton’s method often does not converge close to the dynamic aperture, and sometimes gives misleading results. Moreover, if the symplectification starts with truncated maps that are not exactly order n symplec- tic, the results depend on the way the truncated generating functions are computed. It seems that the general theory provides a good order n symplectification scheme. More about the performance of the algorithm is presented in the next section, where the attention is turned to examples. For a better overview of the theory and aspects of the implementation, the Figure below shows a flow diagram, which explains the algorithmic steps involved in the symplectic approximation / tracking process. ] 1. Compute order n truncation of the map Mn 7 ] 2. Choose a symmetric matrix S, by ithis fixing the generator type utilized ] [ 3. Use (2.9) to compute the order n + ftruncation of the generator of type [S] [ ] 4. Solve (3.175) by iterations, and rise (3.176) to obtain the final result ] r 5. Iterate 4. until the desired number of turns is reached ] We mention that every step is efficiently implemented in COSY INFINITY, and by vectorization of the algorithm, many particles can be tracked simultaneously. Choos- ing different generator types, by specifying different symmetric matrices S, the track- ing algorithm will give different tracking pictures. 72 Chapter 4 Examples This chapter is devoted to illustrate the performance of the extended generating func- tion symplectification method through several examples. First we generate symplectic maps from random Hamiltonians. In two dimensions this can be done easily to high orders. We can assume that these high order truncated maps are approximating the exact maps well enough to be considered numerically symplectic over a sizable phase space region. Then we can compute the generating functions, truncated as some mod- est order (say 7 or 11), and use them to generate exactly symplectic maps according to the above symplectification procedure. Finally, we can compare the various maps obtained this way. We will present some typical cases. Next, we study two examples that have been studied previously in the symplec- tification literature: an anharmonic oscillator [16, 14], and an exactly symplectic quadratic map [51]. These are important cases, as the exact solutions are known and can be compared with the symplectified ones. We apply our method to a lattice of the proposed Neutrino Factory [52], and the FN AL Proton Driver [53]. Although we track the muons for their lifetime (only 1000 turns), it is still an interesting case due to the wide array of nonlinear effects which the lattice exhibits [11, 12, 13]. Finally, some tracking results for the LHC are also 73 provided. 4.1 Maps Generated from Random Hamiltonians We generate polynomials in two variables, starting with quadratic terms. The co- efficients are chosen randomly, evenly distributed in [—1,1]. Regarding the random functions as Hamiltonians, we compute their time 1 maps to very high order, say 19. The vanishing linear part of the Hamiltonians guarantee that the resulting maps will be origin preserving. We show two dimensional examples because the maps can be computed to very high orders, and generically in four or more dimensions the symplectic maps are unstable. The few stable cases that we obtained were linearly coupled and chaotic. As it is well known, chaos cannot happen in time independent one degree of freedom Hamiltonian systems. Although in 2D symplecticity is equiv- alent to area preservation, we chose to Show some 2D examples because due to their regular features it is easier to compare the various tracking pictures involved. At order 19 most of the resulting maps can be considered numerically symplectic over some region of phase space. Thus, we take them as the “exact” results. Then, we compute their generating function truncated at some lower order, and we use them to generate exactly symplectic maps according to the symplectification methods of the previous chapters. We use different types of generating functions, and finally we compare the resulting maps. For one of the random seeds, the 19th order Taylor map tracking picture for 1000 turns, of some particles launched along the q axis, looks as in Figure 4.1 (the coor- dinate axes are always q and p). The symplectified map from the order 19th Taylor map with S = 0 looks almost identical; see Figure 4.2. To asses the performance of the symplectification method for this example, we show the 3rd, 7th, and 11th order 74 0.6 Figure 4.1: 1000 turn tracking of a random two dimensional symplectic map with the 19th order Taylor map (considered to be essentially the exact result). Taylor maps, and the corresponding symplectified maps with S = 0 tracking pictures in Figure 4.3. While the low order Taylor maps give very poor results, the symplectified maps show the right qualitative behavior right from the beginning. The dynamic aperture is overestimated, but the agreement gets better as the order is cranked up, and at order 11 we get almost the same picture as with the 19th order tracking, except perhaps slightly changed tunes of some outer particles. Therefore, at least in this case an 11th order symplectified map predicts the right dynamic aperture. We can compare the 11th order S = 0 symplectified tracking picture with the conventional (Goldstein) generator types. We track using 6 different 11th order gen- erators: F1, F2, F3, and F4 for the full map, and F2 and F3 for the nonlinear part only. These are the cases traditionally used in the past. The results are depicted in Figure 4.4. The superiority of the S = 0 method is clear. The above random map turned out to be the most nonlinear seed. As a second 75 Figure 4.2: 1000 turn tracking of the random two dimensional symplectic map ob- tained from the 19th order Taylor map, by symplectifying it with the generator type associated with the S = 0 symmetric matrix. example in this group, we Show results obtained for another seed, which is much less nonlinear. First of all, the 19th order Taylor map tracking for 1000 turns of some particles, launched with vanishing momenta, is shown in Figure 4.5. Actually, already the 15th order Taylor map gives visually identical results, so we assume the order 19th map to be the exact map. As Figure 4.6 shows, the 19th order S = 0 symplectified map tracking gives the same result as the corresponding Taylor map, even for the tunes. Of course, the lower order symplectifications shown in Figure 4.6 give slightly worse results, but still acceptable, and give a quite accurate estimation of the dynamic aperture. For comparison, the results using the conventional generators are presented in Figure 4.7. We also studied many other seeds for random map generation, and all give similar results. 76 Figure 4.3: 1000 turn tracking of the random two dimensional symplectic map with the 3, 7, and 11th order Taylor maps, and S = 0 symplectified map, respectively. 77 Figure 4.4: 1000 turn tracking of the random two dimensional symplectic map with the symplectified maps utilizing the F1, F2, F3, F4 generating function types for the full 11th order Taylor map, and F2 and F3 for the nonlinear part of the 11th order Taylor map respectively. 78 0.27 Figure 4.5: 1000 turn tracking of another random two dimensional symplectic map with the 19th order Taylor map (considered to be essentially the exact result). 0.27 0.27 Figure 4.6: 1000 turn tracking of the second random two dimensional symplectic map with the 3, 7, 11, and 19th order S = 0 symplectified maps respectively. 79 0.27 0.27 Figure 4.7: 1000 turn tracking of the second random two dimensional symplectic map with the symplectified maps using the conventional generator types F1, F2, F3, and F4 respectively. 80 4.2 An Anharmonic Oscillator We consider the 2D anharmonic oscillator described by the Hamiltonian 1 H : (p2 + q2) — 4(14’ (4.1) [\Dli—a which has been studied previously in [16] to compare various symplectification meth- ods, and in [14] to study Optimal Cremona symplectification. To make the comparison easier we follow the same guidelines, and present the performance of our method by symplectifying the order 3, 7 and 11 Taylor maps of the time 1 map of the flow of (4.1). We track for 1000 turns and use as initial conditions { q = 0.1, 0.3, 0.5, 0.7, 0.9, 0.95, 099, TO , (4.2) p = 0 We present the order 19 Taylor map for comparison purposes. Figure 4.8 shows the orders 19 and 7 non—symplectified Taylor map tracking, and Figure 4.9 the S = 0 symplectified tracking pictures obtained by symplectifying the order 3, 7, 11 Taylor maps respectively. First of all, notice that this system is quite nonlinear close to the dynamic aperture. However, the symplectified tracking pictures are similar to the 19th order Taylor maps already from the order 3 symplectification. The agreement is of course better at order 7, and at order 11 also the edges at the hyperbolic fixed points are becoming visible. Moreover, the tunes of the inner particles become more accurate as the symplectification order increases. For completeness, we also show the order 7 symplectic tracking pictures using the conventional generators in Figure 4.10. Notice that F1 is the best conventional generator for this example. In [16] the conventional generator types F1 and F2 are used; F2 after factoring out the linear part, and F1 for the full map, including the linear part. Some unexpected discrepancies have been observed. By the general theory presented in this dissertation, the reason behind it should be clear: the two types 81 Figure 4.8: 1000 turn tracking of a anharmonic oscillator with the 19th (considered to be essentially the exact result) and 7th order Taylor maps. Figure 4.9: 1000 turn tracking of an anharmonic oscillator with the 3, 7, and 11th order S = 0 symplectified maps respectively. 82 0.8 @ 1.0 \R ””””” Figure 4.10: 1000 turn tracking of an anharmonic oscillator with the symplectified maps from the 7th order Taylor map, using the conventional generator types F1, F2, F3, and F4 respectively. of generators fall into different equivalence classes. Hence, they generate different symplectic maps, with different long-term properties. In the same paper it is stated that the two unstable fixed points of the symplectified maps are moved away from (q, p) = (:tl,0), the locations which correspond to the exact solution. As a matter of fact, this is always the case, for any generating function symplectification method. The explanation lies in Hofer’s metric, and a deep connection between fixed points of symplectic maps and critical points of generating functions (see next chapter). As a consequence, if two symplectic maps have the same fixed points, they coincide. But because symplectification can never restore the true solution, the symplectified maps will always have their fixed points moved away from the locations of the exact solution. 83 ----- ........ -.. - . - . "I’ - ..." ‘W-nn'u-t . ‘ ’ Figure 4.11: 1000 turn tracking of an anharmonic oscillator, with two random gener- ating function types associated with symmetric matrices with entries in [-1, 1] and [—10, 10] respectively. In the next step we studied random generator types, i.e. generators associated with randomly generated symmetric matrices S. In general, if the elements of S are chosen to be small, we obtain better results than if we increased the norm of S. For example see Figure 4.11 for two random types with Si) 6 [—1,1] and S),- E [—10, 10] respectively. These figures represent typical results. While one might say that F1 is acceptable for estimating the dynamic aperture, but clearly S = 0 gives better results. So, can we find an even better generator type? We applied the following strategy: by considering the 19th order Taylor map as the exact map, and wanting to improve the agreement between the Taylor map and the symplectified map for the outermost particle (where the discrepancy is the largest, and expecting that this will not spoil the good agreement for the inner particles), starting with the S = 0 type of generator, we fitted the symmetric matrix S to minimize the radial correction introduced by the symplectification to the trajectory of the last particle, after one turn. We obtained the following symmetric matrix: S, = < 001 _8.4 ). (4.3) The corresponding tracking results are displayed in Figure 4.12. Comparing the various symplectic trackings with the order 19 Taylor map, we see that apparently the 84 Figure 4.12: 1000 turn tracking of an anharmonic oscillator, with the generating function type associated with the symmetric matrix 8,, which was fitted to give the best results. generator based on Sb is the best one, followed closely by the type associated to S = 0. Notice that the separatrix is very well reproduced, and indeed, the excellent agreement for the inner particles is not spoiled. Also, the tunes are predicted accurately over a large phase space region. Therefore, at least for this example, order 7 seems to be enough to estimate the dynamic aperture, if we use the best type of generating function symplectification. It is worthwhile to note that by fitting S using different criteria (such as minimize radial distance over more than one turn and / or more than one particle simultaneously) we get slightly different results. In fact, only Sb (1, 1) is somewhat sensitive to these criteria (up to approximately 10% of its magnitude), but overall the tracking pictures using the different matrices look identical, or almost identical. Moreover, we start fitting these matrices from S = 0, and stop at the first minimum. Thus, we do not know whether Sb corresponds to a local or a global minimum. But since S = 0 is already a pretty good choice for an initial guess, we doubt that there exist better 85 /\ \/ Figure 4.13: 2000 turn tracking of an anharmonic oscillator, with the generating function type associated with the symmetric matrix S = 0, using the 11th order 1/2 time Taylor map. choices of symmetric matrices than 5,. The answer to why is S, the better than S = 0 for this specific case will be given in the next chapter on optimal symplectification. The high degree of nonlinearity close to the separatrix plays a significant role in the outcome of the various symplectifications. Naturally, we can compute the time 1 / 2 map of the same Hamiltonian flow, track it for 2000 turns, and plot every second turn. As a result, we track the same system for the same amount of time, hence the tracking pictures should look the same. However, the maps are less nonlinear, and hence the domain of definition of the generators should enlarge. We did just that with the S = 0 type and order 11, and obtained Figure 4.13. Now the results are much better, and a closer look reveals that the 11th order time 1/2 map symplectified with S = 0 is better than the 11th order time 1 or even time 1 / 2 map symplectified With 31,. As a side note, we mention that the square root trick could be used to gain more confidence in the accuracy of our symplectified maps. Namely, if we track with a 86 symplectified map N turns (N z 103) and plot every turn, then track 2 - N turns with its square root and plot every second turn, we should get the same pictures. If not, we keep taking square roots until the pictures become almost identical. Then, the resulting map can be more or less trusted for long term tracking (N > 103). Finally, to get a better feeling of what the symplectification does at the local level, we plot the corrections introduced by the symplectification method to the tra- jectory of the stable particles over one turn. For example, Figures 4.14, 4.15, and 4.16 show the corrections (magnified by a certain factor for convenient viewing) of the F,, i = 1, 2,3,4 and S = 0 types to the order 7th order Taylor approximation. The spikes point in the direction of the correction, and their length is proportional to the magnitude of the correction. We used color coding to express correction radially outwards (lighter spikes; green in the pdf file), or radially inwards (darker spikes; red in the pdf file). The three initial conditions for the three sets of pictures are (q,p) 2: (0.1,0);(0.5,0) ; (0.9, 0). The general conclusion that can be drawn from these pictures is that if the Taylor map is accurate enough, the best symplectification method introduces the least amount of correction, in both radial and angular direc- tion. Moreover, the corrections in the radial direction are usually much smaller than the correction in the angular direction (the actual values are shown in the upper part of the figures in the following order: number of particles, average radial correction, and average angular correction). This suggests that for dynamic aperture (i.e. region of stability) estimation there is a slightly lower precision needed than for the accu- rate prediction of the tunes (i.e. average angle advances per turn). As expected, the corrections’ magnitude are increasing with distance from the origin and number of turns. In the introduction (chapter 1), it was mentioned that one of the favorable prop- erties of the symplectic methods is their linear global error propagation, in contrast 87 1000 0.3E—3 0 8 0.19E—4 1000 0.52E—3 0.58E-4 _\ § _.._ 1000 0.52E—3 0.58E-4 1000 0.57E—3 0.71E—4 0 8 0.8 1000 0.81E—6 0.24E—9 Figure 4.14: Corrections introduced by the symplectification to particles of small amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = 0 to be on a invariant curve (a stable particle). The figure shows the correction with respect to the 7th order Taylor map of the symplectifications using the convential (the magnification factor is 103) as well as the S = 0 (the magnification factor is 106) generator types. 88 1000 1000 1000 0.27E—2 0.93E—4 0.8 Figure 4.15: Corrections introduced by the symplectification to particles of medium amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = 0 to be on a invariant curve (a stable particle). The figure shows the correction with respect to the 7th order Taylor map of the symplectifications using the convential (the magnification factor is 1) as well as the S = 0 (the magnification factor is 102) generator types. 89 5., 0.041 1000 0.147 o.8 1000 0.190 0.89E—2 Figure 4.16: Corrections introduced by the symplectification to particles of large amplitude, which are predicted by symplectic tracking with generating function type associated with the symmetric matrix S = 0 to be on a invariant curve (a stable Particle). The figure shows the correction with respect to the 7th order Taylor map 0f the symplectifications using the convential (the magnification factor is 1) as well as the S = 0 (the magnification factor is 10) generator types. with quadratic error propagation of nonsymplectic methods. To illustrate quantita- tively this phenomenon, we tracked a particle, with initial condition (q, p) = (0.5, 0), 1000 turns through the time one map of the anharmonic oscillator’s flow, using the exact solution (represented by the 19th order Taylor map), the 7th order Taylor map, and symplectic tracking with the order 7 Taylor map and generator type associated with S = 0. As shown in Figure 4.17, for a few tens of turns the Taylor map tracking is more accurate than the symplectic tracking, then the error of the nonsymplectic tracking grows fast, while the symplectic tracking stays at small error levels through- out the 1000 turns. The linear global error propagation of the symplectic method versus quadratic error propagation of the nonsymplectic method is clear from Figure 4.18, where we superimposed two lines with slopes one and two, respectively, for easy identification. Interestingly enough, initially, for a short period of time, the nonsym- plectic method is accurate enough to pass as a “pseudo—symplectic” one by having slope one, but after z 50 turns the slope suddenly becomes approximately two. 4.3 An Exactly Symplectic Quadratic Map In [51] the following quadratic map is considered: M =NO£, (4.4) Where cos6 sinO £_ ( —sin0 c086 )’ (4'5) With 9 = g, and -3(q+p)2 N( ‘1 ) = ( q . 4.5 p 19+3(q+p)2 ( ) It is easy to check that it is exactly symplectic. We study this map because it is a Simple map with a nontrivial behavior under iteration, and it is exactly symplec- 91 Figure 4.17: Global error propagation in the case of nonsymplectic versus symplectic tracking of a particle for 1000 turns of the anharmonic oscillator’s time one map. For the tracking, the order 7 Taylor map was used for nonsymplectic tracking, and generator of type associated with S = 0 for the symplectic tracking. After a short transient period, the nonsymplectic method gives large errors, while the symplectic method’s error stays small. Figure 4.18: The global error propagation in a log-log scale of the case shown in Figure 4.17. After a short transient period, the nonsymplectic method’s slope is two, and the symplectic method’s slope is one. The dotted lines have exact slopes one and W“), respectively. 92 Figure 4.19: 1000 turn tracking of a quadratic symplectic map with the 3rd order Taylor map (the exact result), and the 3rd order S = 0 symplectified map (which also gives the exact result). tic. Moreover, in [51] it is shown that symplectification of (4.4) using conventional generating functions gives very poor results. As in [51], we also use third order sym- plectification. The exact result and the S = 0 symplectified result for some choice of initial conditions is shown in Figure 4.19. The agreement is excellent. In fact, the results coincide; S = 0 gives the exact solution. As a comparison, Figure 4.20 shows the results for the Goldstein generators, and indeed they give very poor results. The generating function of type S = 0 for (4.4) computed according to (2.9) is F (9.10) = 0-04903810567665787'q3—0.5490381056766573-q2p +2.049033105575557 - qp2 — 2.549033105575553 . p". (4.7) Moreover, it can be shown that the S = 0 generator type can represent exactly any quadratic symplectic map. We also mention that there exist other generator types that can be used to represent exactly any quadratic symplectic map. However, in this group there are none from the conventional, or even the Poincaré types. It was mentioned in section 3.5 that the numerics used for solving the implicit e 1 . . 4 L ‘ ..w— -‘~ 0- ‘- \. . .. O o 004 ,,l"'— ;‘:'._ :.- .-- . - — . .- - ”'7' -.” ' ‘. > 4' " "‘" ....c"""' ‘W'T~~."_ '- ' .7 s I; /':I ’1’ ' " ‘91.» .. N ‘ ‘7'! j": M. _' ' ‘\ NV > '2' "an/'1 V P \\‘\ "\ \A N‘ .r -- l 0 . 002 Jr" 1,! I] \1\ '.‘ 7; I I 4 . » y .7 , ,2 [ \, . ‘ I ' . T‘ I, l l 3 ‘ f O O .4 T. b T. 1 T 1 1 I § 1 it #g I l k. i I. ’ f l ‘ , \ . l I ; [ ‘ ' ._ \ ]> / I" 1' ’ 1 .‘ ,3. ‘.‘V \ 0 , .i ‘ ,1 ’1' I: ., . I. _' ' 1 —0.002 \ \ ,I .I 3, . 7' ‘ . ~7x » I. I “ '1 [ \ ‘ ‘\I:\\“\ -[ , ’ H, F /' .72" 1 . ~;\ ~11: :7 :1: T. I -0.004 "* .., (:1 -- . 7 P ‘5 ‘fi' '5 -. ’“,,.._._- ""' .d . ‘ f i' 4 L i - 15 - 1 - 05 0 0 O . O 5 0 1 0 15 Figure 4.28: 1000 turn (:5 —— a) tracking of the FN AL Proton Driver with the element- by-element numerical integration. the simulation (for more information on accurate fringe field maps for the LHC see chapter 10) are shown in Figure 4.30. For the sake of completeness, wedepict the results without and with fringe fields for the case of F2 and F3 symplectified nonlinear parts of the map in Figures 4.31 and 4.32. Again, the terrible performance of some of the generator types point out the necessity of optimal symplectification studies, and shows that the right generator type (S = 0) gives excellent results once again. The topic of optimal symplectification from a very general standpoint is the sub— ject of the next chapter. 102 0.002 0.002 (a) 8th order Taylor map (b) S = 0 symplectified map 0.002 0.002 (c) F1 symplectified full map ((1) F2 symplectified full map 0.002 0.002 (e) F3 symplectified full map (f) F4 symplectified full map Figure 4.29: 105 turn tracking of the LHC with the 8th order Taylor map, and the Corres.ponding various symplectified maps, without fringe fields taken into account. 103 0.0003 0.0003 __ - .3 ............ M __ (a) 8th order Taylor map (b) S = 0 symplectified map 0.0003 0.0003 __ __ ................ _Q__._O_Q5 (c) F1 symplectified full map ((1) F2 symplectified full map 0.0003 0.0003 M ................ _Q_._Q_Q_5 __ ................ _O__O_Q_5 (e) F3 symplectified full map (f) F4 symplectified full map Fig‘lre 4.30: 105 turn tracking of the LHC with the 8th order Taylor map, and e Corresponding various symplectified maps, with detailed fringe fields taken into aCCGunt. 104 0.002 0.002 (3) F2 symplectified nonlinear part of the (b) F3 symplectified nonlinear part of the map map Figure 4.31: 105 turn tracking of the LHC with the F2 and F3 symplectified nonlinear parts of the 8th order Taylor map, without fringe fields taken into account. 0.0003 0.0003 (3.) F2 symplectified nonlinear part of the (b) F3 symplectified nonlinear part of the map map Figure 4.32: 105 turn tracking of the LHC with the F2 and F3 symplectified nonlinear parts of the 8th order Taler map, with detailed fringe fields taken into account. 105 Chapter 5 Optimal Symplectification Among many systems of practical interest, hadron colliders in the single particle approximation can be modeled as Hamiltonian systems. Hamiltonian systems are uniquely characterized by their symplecticity [10]. One of the fundamental quantities in accelerator physics is the dynamic aperture (DA), which, roughly, is the region of space containing stable particle orbits over long times. Since the system is so complex that an exact solution is not within reach, simulations are needed to estimate the DA [13, 56]. This can be achieved by iteration of the so—called one-turn map, i.e. Poincaré section map, of the system. Unfortunately, only some approximation of the one-turn map, as, for example, the order n truncation of its Taylor series, is available [24]. While the Taylor map preserves the symplecticity up to order n terms in the expansion, in general fails to be exactly symplectic. The numerical simulations in chapter 4 show that the truncation often generates inaccurate results. Therefore, restoration of the exact symplecticity of the one-turn map is desirable. There are several symplectification methods [14, 15, 57]. While every method produces exactly symplectic maps, the results are not equivalent. The symplectified maps depend on the specifics of the methods. Several examples are presented in chap- ter 4 using the formalism of generating functions of canonical transformations. For 106 some generators, the results are not satisfactory. Therefore, it is not only important to symplectify, but also to symplectify the right way. The purpose of this chapter is to give a precise meaning for how to symplectify “the right way” [58]. 5.1 Formulation of the Problem As with any approximation method, a criterion for closeness is needed; mathemati- cally speaking, a suitable metric is necessary. In our case, the metric should provide a way to measure distances between Hamiltonian symplectic maps, and should have some desirable properties, namely: 1. The symplectification should work well for every particle in a given Poincaré section, 2. The outcome of the symplectification should not depend on the specific Poincaré section used, 3. The symplectification should work just as well after any N > 1 turns as after one turn, and 4. Based on the previous three conditions, the assessment of the optimality of the symplectification should be unambiguous. These conditions can be captured by the requirement that if a symplectification method yields the best result, say M, with respect to the metric, then the same Symplectification gives the same result for A o M o A'l, A being any symplectic map, which entails coordinate independence of the metric. It is hOped that such Special purpose metrics would capture better the details of the dynamics than general Purpose metrics (as, for example, the well-known C0 metric), and would give an 107 as unambiguous as possible way to measure distances. Therefore, mathematically speaking, we would like to have a bi-invariant metric for Hamiltonian symplectic maps. The importance of bi-invariant metrics has been pointed out also in [59]. It also can be thought of as a certain generalization of subsection 3.4.1, where it has been shown that linear symplectic variable changes do not matter, to the nonlinear case. The question is whether such a metric exists at all. Indeed, existence of such a metric is highly nontrivial over an infinite dimensional non-compact Lie group, like the Lie group of symplectic maps. A negative example from the field of motion planning for robotic systems is provided by [60], where it is shown that over the relevant Lie group, i.e. SE (3), no such “natural and unequivocal concept of distance” exists. Hence the results are task or designer biased. 5.2 Hofer’s Metric and the Optimality Condition Fortunately, there exists an outstanding metric that satisfies our needs, and sym- plectic topology provides a way to formulate the necessary conditions for optimal symplectification. This is astonishing, since despite Hamiltonian systems have been studied for such a long time, there was no symplectic topology 25 years ago. Now symplectic topology is a very lively research field, and we will use and extend some results concerning Hofer’s metric. In [18], a surprising intrinsic metric has been intro- duced, now called Hofer’s metric, on the space of compactly supported Hamiltonian symplectomorphisms, Hamc (R2’T). Recall that a symplectic map is called Hamilto- nian if it is the time one map of the flow of some function defined on phase space. The fact that such a Finsler metric exists on a non-compact infinite dimensional Lie group points out the special nature of Hamiltonian systems. 108 5.2.1 Hofer’s Metric We give a short description of this norm [45]. In general, let G be a Lie group with Lie algebra L. A norm [I . [l on L is called invariant if it is invariant under the adjoint action of C ll 6 l|=|| g‘lég II (51) (defined through the exponential of I; at t = 0) for every E 6 L and every 9 E G. Any such norm gives rise to a bi-invariant intrinsic metric on the Lie group via 400.9.) = if] H 50mm“ 1 dt (52) for g0, gl 6 G. The infimum is taken over any smooth path g : [0, 1] —> G connecting 90 = 9(0) 130 91: 9(1)- Specifically, the compactly supported Hamiltonian vector fields of R2" can be iden- tified with the space of compactly supported functions Cf" (112") via the isomorphism mentioned in subsection 2.2.1. The velocity vector of (5.2) is the Hamiltonian vec- tor field. Hence, Hofer defined the following L°°-type norm on compactly supported Hamiltonian functions [lXHtH é sup Ht (2) — inf Ht (2) (5.3) z€R2n zeRZn For convenience we denote it as ||Ht||. This norm is called the oscillation norm. The adjoint actions are the symplectic variable changes. For the Hamiltonian functions the adjoint actions are the transformations H +———> H o w, for every H E Cf,’o (1112") and every w E Symp (132", J). The oscillation norm is obviously invariant under the ad joint action ”Ht“ = [IHtOTDH (5-4) 109 In fact, the norm stays invariant under the larger diffeomorphism group of R2". The induced length spectrum for paths {55,}, t E [0, 1] in Hamc (1132") is given by 414.}: / “Hindi: (5.5) where H, is the, possibly time dependent, generating Hamiltonian. For any two 90, 1/1 E Hamc (1112"), the distance between them is defined as 1 72(52):, inf 414.}: inf ,/ llHtlldt (5.6) 0 50:53:17 (907-39451: The infimum is taken over all smooth paths in Hamc (R2n) from cp to 7,1). The following proposition holds [46]: Proposition 9 For all 59,90,112 E Hamc (Rm) the following hold: 0 72025.90) 2 0, p(¢.¢)=p(ow,9oow) the map t —-> p(qb,,I) is uniformly continuous. The first three properties mean that p is a pseudo-metric. The highly nontrivial fact is [46]: Theorem 10 Let 05,99 6 Hamc (132"). Then p(d>,90)=0 ¢> ¢=

0 Furthermore, L = Jac (L) E Sp (272,13), and it is well-known that Sp (2n,R) is con- tractible. There are several ways to see this. For example, any L E Sp (2n, R) can be written as L = eJSIeJST, (5.12) Where 31,52 are symmetric matrices [62]. Therefore, it is enough to define the arc £1 = 6‘351 o BUST, (5.13) t E [0,1] to obtain the final part of the total isotopy. Putting together the different parts by juxtaposition of paths, we obtain the following piecewise smooth isotopy: Lt, te [0,1/3], M, = M2, tE (1/3,2/3], (5.14) tc+M°, tE (2/3,1]. 113 By replacing the parameter t by a smooth function f : [0.1] —+ [0,1] such that it is constant in the neighborhoods of the non-smooth points, we finally obtain a smooth isotopy from the identity to M, showing that every M E Symp (R271) is in fact E Ham (R2'T). The next natural question is the following: if M E Sympc (1112"), then is it true that M E Hamc (Rim)? In general, the answer is not known. It is true for n < 3 [50]. Moreover, it is known that Hamc (1132") is the C0 closure of Symp“ (Rm) for any n. Again, there are several ways to see this. Perhaps the easiest is to notice that for any compactly supported diffeomorphism 55 such that supp(¢) C D, and any diffeomorphism 6, supp(0 0 ()5 0 6‘1) C 0 (D). It follows that any M E Sympc (Rm) can be conformally rescaled to have support in an arbitrary small neighborhood of a point. Then, the Alexander trick (5.10) gives an isotopy from M to an element in Sympc (182") arbitrarily Co-close to the identity. Therefore, for computational pur- poses we can always interchange symplectic with Hamiltonian, even in the compactly supported case. 5.2.3 The Optimality Condition To this end, Hofer’s metric can be applied to the problems of interest to us. There— fore, optimal symplectification can be defined as the symplectification method that minimizes the distance in Hofer’s metric between the exact map and the symplectified maps. That is, if the set of all possible symplectification methods is denoted by E, the best result is achieved by any symplectic map New which satisfies p(MaN0pt) = 1gp (MA/7) - (5-15) While being very general, there is a problem with this formulation of the optimal symplectification, namely it is not very useful for practical computations. The reason 114 is that in general it is not known yet how to compute the Hofer distance between two arbitrary maps in Hamc (1132" ). The difficulty lies in the necessity of consideration of all the Hamiltonians generating the two maps, or equivalently, the paths in Hamc (Rzn) from M to M. However, by the nature of our optimality condition, we are interested only in the maps N;- that are already close to M in some sense. Obviously, this necessary condition can be achieved by sufficiently increasing the value of n, the degree of the polynomials M, with which the exact maps are initially approximated. Thus it would be sufficient if a suitable neighborhood of M can be parametrized in such a way that (5.15) becomes computable. Clearly, any symplectic map that does not fit into this neighborhood cannot be optimal. 5 . 3 Link to the Extended Generating Function The- ory Indeed, parametrization of a neighborhood of M is possible in the C1 topology, uti- lizing the theory of generating functions. The first results in this direction have been Obtained in [63] for Hamiltonian maps C1 close to identity and Poincaré’s generating function, and then it was extended to Hamiltonian maps C1 close to identity and all compactly supported generating functions in [64] and [45]. While the approach of [64] is more general, as it holds on any symplectic manifold, we are only inter- ested in R2“, and the method of [63] lends itself more easily to generalizations. The main idea is to replace the Hamiltonian maps by their generating functions, and try to BXpress Hofer’s metric between two maps as some norm of the difference of their generating functions. In [63] this was proven to be possible in some cases. However, in the extended theory of generating functions of chapter 2 it was shown that in fact there exist uncountably many generator types for any symplectic map, some of which are not compactly supported. To be able to decide which generating function type 115 provides the optimal symplectification, the result of [63] must be generalized to every generator type. More precisely, introducing a map (1),, that sends a symplectic map M into its generating function F of type a, we prove that (Do, is an isometry. Formally, we can state the result as the following theorem: Theorem 11 There exists a neighborhood 8 of any M E Ham“ (RM), and a neigh- borhood Z ofO in 03° (R2n) such that the map (1)0:8 —> 3 CD0 (M) = F (5.16) is isometric. That is, for every F, G E Z, 1 1 p(M,N)=—Il‘1>a(M)-0(N)ll=—||F-G||o (5-17) [It] lul As a consequence, the inverse of the isometry takes any function from (Z, l] - ll) into a Hamiltonian symplectomorphism in (E, p) depending on a. This shows that, considering the space (Z, I] - H) a flat space (in which straight lines are minimal geodesics), their image under (1);,1 : Z —-> E in HamC (R2n) remain locally flat min- imal geodesics. That is why this results is called the local flatness phenomenon. The proof uses three main ingredients. First, there is an intimate relationship between fixed points of symplectic maps and critical points of generating functions, which is presented in the next subsection. Second, the proof in [63] is based on the Hamilton—Jacobi equation. In subsection 5.3.2 the generalized Hamilton-Jacobi is derived, adapted to our situation. It provides the time evolution of any generator type. Finally, the proof uses a theorem of Siburg [65], which is stated without proof in subsection 5.3.3. The theorem essentially states that paths without nontrivial fixed points are absolutely length minimizing for Hofer’s metric. 116 5.3.1 The Fixed Point-Critical Point Relationship In this subsection we study an interesting property of generating functions. A point Zf is called a fixed point of the symplectic map M if it acts on it as the identity map, that is M (zf) = 2;. It follows that the iterates of the map have the same fixed points, Mk (2,) = Mk“1 0 M (2,) = Mk"1 (2;) = = M (2,) = 2,. (5.18) Moreover, the inverse M‘1 also has the same fixed points, M"1(zf) = M‘1 o M (2f) 2 zf. (5.19) To sum up, we can say that any integer power of a symplectic map has the same fixed points as the map itself. Thus the set of fixed points form a topological invariant of the map under iteration. Generating functions can be connected with these fixed points. For start, let us consider a special class of generating functions. We assume that the generating functions are globally defined, otherwise the theory is valid for the fixed points in the regions where it is defined. The critical points of functions in this set are the fixed points of the symplectic maps, and conversely, the fixed points of the map are critical points of the function. If the symplectic maps are compactly supported, then these generating functions are exactly the generating functions with compact support. This can be easily seen if we argue geometrically. Recall that the symplectic map is a Lagrangian submanifold in the product manifold. Then, the fixed points of the map are exactly the intersection points of this submanifold with the diagonal. On the other hand, the critical points of the generating function are the intersection points of the Lagrangian submanifold determined by the function with the zero section. Now, if we identify the diagonal with the zero section by a, the fixed points of the map will go into critical points of the generating functions. 117 Explicitly, given a symplectic map, we can choose only as that satisfy the transver- sality condition. On the other hand, identification of the diagonal with the zero section requires that a (A) = Z, (5.20) which further restricts the pool of as. A necessary condition is that the generating functions in this class can generate the identity map. If the map is close enough to the identity, then also the transversality condition is satisfied automatically. Equation (5.20) can be expanded as (11 O (5.21) NNHH E II A NO V E (120 Applying Zf to (2.107), we obtain that VFoazo(“‘I")(z,)=alo(AI")(z,), (5.22) which is equivalent to VFoazo(§)(zf)=ozlo(§)(zf). (5.23) Using (5.21) we arrive to VF(wf) = , (5.24) where w f is given by 2f and the identification process. From the explicit constraints we will see that actually to; = 2f. Hence, the fixed points of the map are critical points of the generating functions from this set. The next question is whether this is true conversely: are all the critical points of generating functions in this set fixed points of the map? As has been shown in section 118 2.4, M=(olo(VZF ))o(a2o(VIF ))_l, (5.25) Mo [52 o ( VIF )) (we) = (51 o ( VIF )) (we), (5.26) where we are the critical points of F. From (5.20) it follows that 01‘ 0’1 (Z) = A, (5.27) (m=(§)e) as) NoNo Combining what we have derived sofar, we are able to answer the question about critical points. Using VF (me) = 0, from (5.26) and (5.28) we obtain M (2c) = 26. (5.29) Again, 26 is the point corresponding to we via the identification, and we will see that 2:C = we. Hence, we are fixed points of the symplectic map. In conclusion, there is a one-to—one correspondence between fixed points of the map and critical points of the generating functions in this set. Now we elaborate in the direction of finding the explicit constraints for this class of generating functions. Beside the constraints (2123-2125) we have another set given by (5.21), which in terms of the entries in the Jacobian of a read A (z, z) + B (2, z) = 0, (5.30) C (z, z) + D (2, z) = I. (5.31) 119 But we already know from chapter 3 that, with M = I, this set is nothing else than generating functions constructed via any a of the form _ —aJ a] MGR." a_(%(I+JS) %(I—JS)) 5:57“ (5°32) The critical points of these generators are in one-to-one correspondence with the fixed points of the map. Now it is straightforward to check that for any generating function in this set indeed w f = Zf and 2C 2 we. The above results can be extended to any generating function, with the condition that it can be used to generate the identity map. The difference between the above set and the other generating functions is that they will not identify the diagonal with the zero section, but some other section. However, this little inconvenience can be circumvented by noticing that subtracting from the generating function the function that generates the identity map, we get almost the same results. That is, suppose we have the following: VF5525(’¥)(2)=515(AZ")(2), (5.33) vrooago(§)(z)=alo(§)(z), (5.34) where we denoted by F0 the same type of generating function as F, but which gen- erates the identity map. Then, if z = 2;, the right hand sides are the same, and we obtain that V (F — F0) (wf) = 0. (5.35) The fixed points can be obtained from 2, = (520 ( g D4013). (5.35) 120 So, there is still a one—to—one correspondence between fixed points of symplectic maps and critical points of generating functions in the above modified sense. Of course, in the case of the (5.32) set F0 = 0, and the previous results can be recovered. Finally, to put the results in the form that we will need them to prove the theorem, consider the case when the same generator type a exists for two symplectic maps M and N. It follows that vroa2o(/‘I")(z):a,o(";)(z), (5.37) “If VGoago(';/)(z)=alo< )(z). (5.38) Then, on the set of common fixed points Zf of M and N we obtain that v (F — o) (w,) = o. (5.39) 5.3.2 The Generalized Hamilton-Jacobi Equation and Appli- cations In this subsection we prove the most general Hamilton-Jacobi equation, associated to the time evolution of any generating function. As we have seen, critical points of generating functions are closely related to fixed points of the map. But more can be said. We will show that, actually every generating function of compact support assumes the same value at the fixed points of the map. These numbers are called the action of the fixed points, that is, if z, is a fixed point of M, define A (zf, M, ,u) as the value taken by any generating function. Call the set of all such numbers, as 2f spans all the fixed points, the action spectrum of M 0,, (M) = {Ahab/Wu) [M (2;) = Zf}- (5-40) The spectrum depends parametrically on the conformality factor a of a, and is a symplectic invariant of the map. To show the claim, we need to prove that the 121 generating functions satisfy a generalized Hamilton-Jacobi equation. Based on the proof in [28] for linear as, here we present the general proof. Assume that the flow is generated by the time-dependent Hamiltonian Ht, and the corresponding time t maps can be represented at any moment by a time-dependent generating function Ft. We claim the following. Theorem 12 The following Hamilton-Jacobi equation is satisfied for any to E R2": giF‘ (w) = “H, o (0.1 o ( A14 )) (7.0), (5.41) where Ft is the generating function associated to any a satisfying (2123-2125), and M = (VFt)T at every moment t. Proof. We are situated in the (Rm, J4") symplectic space, with symplectic coordi- nates (w, 122). Consider the extended phase space RM” by including the canonical pair of variables (t, —Ht). The symplectic structure modifies to w = (410 — (It /\ dHt, (5.42) the differential form of Cartan. Take the one-form e = IL’d'lU + tht, (543) where H, = “H, o (a1 o ( g? )). We prove that A6 is closed. We have that 2" 62f; 2" so 2" 6H 0H _ __i , ,. . i . _£ . ___‘ dAe _ij=1 awjdu, /\ dw, + 2:; at dt /\ dw, + 2:; 6w.- dw, /\ dt + at dt /\ dt 2" 2" so of; _ 22; (Jac(/Vt) — (Jac(/Vt) )) dw, /\ dui, + 2(61: 3111,) dt /\ dw, 2" (912’ BE '2' t = —— -— dt /\ d' ,-. . .2; ( at ow.) w (5 44) 122 On the other hand du? 61D fit: 2 M (w) = Jac (M (u!)) - u'} + 5;, (5.45) @512 = N, (w) — Jac (N, (15)) .25. (5.46) Recall that w = a2 0 ( A2” ) (z), (5.47) a = a1 0 ( A2“ ) (2), (5.48) and 2 = M, (2) is the solution of the initial value problem d2 . a Z JVgHt (Z), (5.49) 2 (t = 0) = 2. (5.50) We compute 6H, _ ( M \ 5; (w) _ qu (Ht 0 (a1 0 K I )J) (10) (5.51) T = u [Jac (a1 o ( (gt )) (w)] 'VgH (2?) (5.52) . N. T - = u [( A“ B“ )-( I )] ~V2H(z) (5.53) = ,u (AO‘Nt + B“)T - VEH (2) (5.54) = u (NtAaT + BOT) -V,H (2), (5.55) where in the second row we used 2 = a1 o ( ‘9? ) (w), in the third row Jac (M) = N, and the notation for the J acobian of a—1 _ A“ B“ a# = Jac (a 1) = ( C0 D“ ). (5.56) 123 In the last row the fact that N, is symmetric, N,T = N, is also utilized. We can express a# in terms of entries of a#. From (2.113) it follows that 1 .. a# = ——-J4,,a;J4,, (5.57) p _ 1 J2n 02n AT CT O2n I2n — —; < 021: —J2n ) ( BT DT ) ( _12n 02n ’ (5.58) which reads explicitly 1 1 AC! = —J,,,CT, so = ——J2,,AT, (5.59) H H a 1 T a 1 T C = ———J2,,D , D = —-J2,,B . (5.60) Li H Finally, we obtain LH‘ « — v.40T BOT V H * 3w (w) — #(1 t + ) 2 t(z) (5‘61) Analogously, the calculation of ‘33—? proceeds as follows Bu”) (1 d ‘37- : 81(01 (2,2)) —Nt' E(a2 (2,2)) (5.63) _ 8a1(é,z) (12 802 (2,2) d2 — o2 'E‘Nt' 62 '5 (5°64) Hence, dAe = O. The vanishing first cohomology class guarantees the existence of a function F, (112) such that dF, (w) = A, = ide + H, (w) dt, (5.66) (9F, (1U) 3F, (TU) _ A . — . 8w dw + at dt — wdw + H, (w) dt. (5.67) Comparing coefficients, we get that indeed F, (10) is the generating function VF, (w) = M (w) = u}, (5.68) 124 and the advertised result BFng) = th 0 (0,10 ( A; )) (,0) (5.69) Finally, as an application, it can be shown that compactly supported generating functions assume the same value at the fixed points of the symplectic map. Notice that the right hand side of (5.69) can be expressed as H,o (51 o ( A; ”(15): H,o (5'1 oao( Ag“ )(2)) (5.70) = H, o M, (2). (5.71) In the derivation we used the following _ a1 o a aloa=(a2oa)=l. (5.72) Therefore, a1 o a is the identity for the first 2n components and 0 for the second 2n components. Also, if we consider only time-independent Hamiltonians, H is invariant under its own flow at any time, H 2 H o M,, since H is constant along the solutions of the Hamiltonian dynamical system. Reparametrization with respect to time, such that M is the time 1 flow of H,, and integration of (5.66) gives Foa,o(/‘I")(2)—Fooa,o(§)(2)= (5.73) [1:11) VF (w) - dw + u‘/01H,0M,(z)dt. (5.74) We used M0 = I, and the notation F, = F. If 2 = 2;, the first term on the right hand side is vanishing because in this case 111(0) : 10(1) as can be seen from (5.47). Hence, we get (F — F0) (w) = u [0 H. o M. (2,) dt. (5.75) 125 with I w, : a2 0( I > (3,). (5.76) In particular, for compactly supported generating functions (5.75) takes the form 1 F(zf)=u/ HtOMt(Zf)dt=A(Zf,M,/,l). (5.77) o This means that generating functions in this set, with the same conformality factor, take the same value at the fixed points of the symplectic map. In general, the other generating functions, after subtracting the part that generates the identity, take the same value at points that are in one-to—one correspondence with the fixed points. In [45] it is shown that the action of a fixed point is related to the area enclosed by certain loops, thus is no surprise that the action depends parametrically on the conformality factor. It is also known that the action spectrum of compactly supported Hamiltonian maps is compact and nowhere dense, and in general does not depend continuously with respect to M. For details we refer the reader to [46]. On the other hand, Weinstein proved that the fixed points of a perturbed symplectic map are close to the fixed points of the unperturbed map [32]. A related result is due to Viterbo, who proved that any compactly supported symplectic map has infinitely many periodic points inside its support [66]. 5.3.3 Siburg’s Theorem Some preparation is needed before the theorem statement. Definition 3 Let I C R be a connected subset with non-empty interior. A smooth path {¢,} : I —+ HamC (R2") is called regular if XH, # 0 for every t E I. 126 Definition 4 Let {at} : I ——> Hamc (R2n) be a smooth regular path. 0 {(25,} is called a minimal geodesic if for all a, b E I, such that a < b, 5 {‘15:} l[a.b] = 100%» $5) (5-78) holds. 0 {(15,} is called a geodesic if for every t E I there exists a neighborhood U C I of t such that {9b,} [U is a minimal geodesic. The following proposition is proved in [63]. Proposition 13 Let H : [a, b] X R2" ——2 R be a smooth compactly supported function. The following two conditions are equivalent: - ffllHtlldt = Ilfth(z)dtll 0 There exist two points z_, 2+ 6 R2" such that supH, = H, (2+) and ian, = 2 H, (2-), for allt E [a, b]. Definition 5 A function H, (2) which satisfies one of the conditions in the proposi- tion above is called quasi-autonomous. Each autonomous path is quasi-autonomous. We are interested under what conditions the length-minimizing property of a path is achieved. Lalonde and McDuff [64] proved the necessary condition, which holds for any symplectic manifold. Theorem 14 A regular path {45,}, t E I, in HamC (R2") is a geodesic if and only if its generating Hamiltonian has at least one fixed maximum and one fixed minimum at each moment. 127 Hence, every geodesic is generated by quasi-autonomous Hamiltonian functions. To prove that a path is length—minimizing is much harder. Fortunately, in R2" one can obtain more results than in the general case. First, we need another definition. Definition 6 A fixed point, 2f, of a map (151 E HamC (Rin) is called constant if it is a fixed point of its flow, that is 45,251 = 2,, for every t 6 [0,1]. We say that a smooth path {omlo’ll} starting at the identity has no non-constant fixed points if for any fixed r E (0,1] and 2; such that (15421) = 27 => 4% (21) = Z; (5-79) for any t E [0,1]. In case {new} does not start at the identity, (5.79) should be modified to 95. (37) = ¢o (2;) => i9, (Zr) = $0 (Zr) (5-80) We are ready to state the following theorem [65]. Theorem 15 Any regular path {¢te[0,1]} in HamC (R2n) that is generated by a quasi- autonomous Hamiltonian, and has no non-constant fixed points in time less than 1, is a minimal geodesic, that is absolutely length-minimizing for Hofer’s metric. It is conjectured that a similar theorem holds for any symplectic manifold. The case of autonomous Hamiltonians has been proved recently [67]. 5.3.4 Proof of the Main Theorem Now we are ready to derive the so-called local flatness phenomenon. We want to measure the distance between two compactly supported Hamiltonian symplectic maps, say (,0 and 1]). Suppose that for some type a the two generating functions 128 associated with a for go and v are Fa and Go. Take the convex combination of the two generating functions S, = (1 - t) Fa + tGa 2 Fa + t (Ga —- Fa). (5.81) The corresponding path {(15,} in Hamc (R2") is generated by a Hamiltonian H,. The path satisfies the following relations: (to 2 cp and o, = w. Noticing that So = Fa, if z, is a fixed point of (1),, for some fixed 7 E (0, 1], according to (5.39) it follows that V (S, - 50) (w!) = T ' V (Ga - Fa) (wf) = 0, (55-82) that is w, is a critical point of Ga — Fa. Obviously, this implies that w I remains a critical point of Ga — F0 for any t 6 [0,1], which in turn means that z, is a fixed point of o, for any t 6 [0,1]. Indeed, as in (5.36), it follows that in this case 2, 2(a20(§))1(wf). (5.83) According to subsection 5.3.1, we just proved that all the fixed points of {¢,€[0,1]} are constant. Also, a careful look at the Hamilton-Jacobi equation (5.69) reveals that the path {¢te[0,1]} is generated by quasi-autonomous Hamiltonians. Since, as follows from above, the left hand side is quasi-autonomous, the right hand side must be quasi-autonomous too. Moreover, the points where the left hand side achieves its maximum and minimum values are critical points of Ga — Fa; hence at these points M(w,.) = VS,(wC) = VFa (we) is time independent, showing that indeed H, is quasi-autonomous. We remark that the same arguments show that paths with these properties are never unique, as in the definition of S, any function of t such that f (0) == 0 and f (1) = 1 can be taken instead of t. In our specific case, i.e. f (t) = t, in fact autonomous paths are obtained. As a side note, this shows that symplectic maps close to identity always can be generated by autonomous Hamiltonians. 129 To this end, the conditions of theorem (15) are satisfied, and the Hamilton-Jacobi equation provides a method to compute the distance between symplectic maps. Using (5.71) in (5.69), taking norms on both sides of (5.69), and using the invariance of the oscillation norm under the adjoint action, we obtain “Ga - all: lll‘HtOMtll: lttlllHtll- (5-84) Integration gives ”Ga - all = lul 'g{¢t}v (5-85) and according to Siburg’s theorem we finally arrive at 1 p (w. w) = mllGa - Fall. (5-86) In summary, as long as the same type of generating function exists for two com- pactly supported Hamiltonian maps, the Hofer distance between them can be mea- sured as the oscillation norm of the difference between their generators. From our point of View, it is very important that this result holds for any generator type, sub- ject to the existence condition. This is always the case if cp and it) are sufficiently close in the C1 topology. However, it is a local result in the sense that it does not provide a way to compute Hofer’s distance between any two arbitrary Hamiltonian maps. It is worthwhile to note that for any autonomous Hamiltonian H, there exists an e > 0 such that the time 1 flow of 5H has only constant fixed points [45]. Also, it auto- matically follows that flows of quasi-autonomous Hamiltonian systems are geodesics, that is they minimize length on sufficiently small time intervals. It was known from [46] that this is true for autonomous Hamiltonians, and examples of time dependent (quasi-autonomous) Hamiltonians can be found in [68]. 130 5.4 The Best Generating Function Type Now we are ready to transfer the problem of solving (5.15) to solving , 1 10(M’N0pt) = gmllqh (M) - (Pa (M) ||- (587) Denote (D0 (M) = F 0. Unfortunately, F a is unknown, and to minimize the right hand side of (5.87) a good approximation of F a is needed. All the information about the system that is available in practice is contained in M,,. This entails that, with some a priori fixed a, the best approximation for F a is obtained by solving as accurately as possible (2.9). The necessary operations of truncated map computation, map composition and order m inversion, and integration are readily available in the code COSY INFINITY [24], and, as a consequence, the order m Taylor expansion of F 0', F3, can be easilyobtained. Then, it follows that the best result is achieved by the symplectic map Nap, which satisfies 1 0 =' f— a— a . . 10(M2Npt) (IIIEIEI/JlllF Fm“ (5 88) where (1),, (M) = F,‘;',. Apparently, minimization of the right hand side of (5.88) is equivalent to the choice of the generating function type that achieves this minimiza- tion. It is worthwhile to note that, due to the one-to—one correspondence between gen- erating functions of a fixed type and symplectic maps, p (M,No,,,) > 0 always, which means that the true solution can never be recovered by symplectification. Therefore, the differences among symplectification methods is caused by the truncation of the generating functions. However, F,‘,", is the most that can be computed in practice. Hence, we can state the following conclusion: Optimal symplectification with respect to Hofer’s metric is achieved by the order n truncated generating function type that has the smallest oscillation norm of the terms neglected, above order m. 131 Thus, the remaining question is that, based on this limited information, which generator type will give Optimal results in general? To answer this question we need to pick a generator type, or equivalently an a, that minimizes ”Fa—Ff,” /u. Again, for an arbitrary M this turns out to be a difficult problem, because in general nonlinear as would be required, and it is difficult to construct useful nonlinear maps that satisfy (2.6). For accelerator physics applications this turns out not to be a problem since the maps of interest are in general weakly nonlinear maps around equilibrium points. For these types of maps linear choices of a are sufficient, which simplifies the construction of generating functions. Also, in principle any nonlinear Hamiltonian map can be split into a composition of Hamiltonian maps which are only weakly nonlinear. Therefore, the final step is to find the linear a, which can be considered as matrices, such that HF“ — Fall/u is minimized in general. One of the main results of chapter 3 is that the set of linear maps satisfying (2.6) can be organized into equivalence classes, meaning that for symplectification purposes the following are the only independent generator types: _ ~1 0’ 2 ( §(1 5%) M-1 51:15) ) ’ (5'89) where M is the linear part of M, and S represent arbitrary symmetric matrices. For a given M with linear part M, the classes characterized by some symmetric matrix S is denoted by [S]. We note that M is know from M,,, and u = 1 for every a from (5.89). Thus optimal symplectification is map dependent, that is, there is a different optimal symplectification for every symplectic map having a different linear part. Then, which class [S] gives the optimal symplectification for symplectic maps having the same linear part? To answer the question, first is observed that instead of the requirement of minimization of HF" — F3“, minimization of ||F°|| could be required. The two requirements are not strictly equivalent. In one direction there is 132 no problem; if we require F“ to be small, it follows that the tail F“ — Ff; will be also small. Indeed, the only way that F“ — Fr“, will not be small for some m when F“ is small is by cancellation of large terms in the Taylor expansion of F“. However, as it will be seen below, this cannot happen due to the fact that C0 smallness implies C 1 smallness. The other direction is not necessarily always true, that is, in general a small F“ — Ff}, does not imply a small F“. However, as have been shown in subsection 2.2.5, a large F“ does imply in general a small domain of definition of F“. For further studies that support this conclusion, utilizing several examples, see [44]. It was shown that, indeed, lower bounds for the domains of generating functions can be guaranteed tooenclose the dynamic aperture, if the corresponding types are small enough. Since we are interested in estimation of the DA, the generating functions should be defined in a rather large domain in phase space. Putting together the facts, we can conclude that for our purpose it is sufficient if ||F“|| is minimized. To this end, notice that, with the notations 23 = M (2) and w = a2 (:2, z), (2.9) can be expressed as VwFa (w) = a1(2,z). (5.90) Integration, which can be along an arbitrary path according to Stokes’ theorem, gives Fa (w) 2 /0w a1(2,z) -dw'. (5.91) Taking norms on both sides of the equation the following estimate is obtained: nwuwm(?)(Mw(?)i on It is rather straightforward to check from (2.9) and (5.89) that a1(2,z) = 0+0(22), (5.93) a2(2,z) = I-z+%(I+JS)-(’)(z2). (5.94) 133 From these equations it can be inferred that indeed CO smallness implies C1 smallness. Therefore, a, (2, z) is already small if M is weakly nonlinear, and its norm does not depend on the type of generating function. Hence, minimization of [IF “H in the end is equivalent to minimization of I + (I + J S ) / 2 - O (22). The only free parameter is the symmetric matrix S. Because the 0 (22) comes from the nonlinear part of the symplectic map M, a simple calculation shows that S = 0 is the best choice if M is allowed to be free. To see this, denote the Jacobian of M by M + N, where N is the nonlinear part. Because M is symplectic, we have the following relation: NTJN + NTJM + MTJN = 0. (5.95) We also observe that minimization of (5.94) and of 1 [[J(I+§(I+JS)-N) I] (5.96) are equivalent, since J just reorders the entries. Suppose we found some S such that J(I+%(I+JS)-N)=J+$JN—%SN%O. (5.97) Hence we have the relations MTJN z MTSN—2MTJ, (5.98) NTJM z NTSM + 2JM, (5.99) which inserted into (5.95) gives [NTJN] + [MTSN + NTSM] + [2.1M — 2MTJ] z 0. (5.100) We separated the terms into three groups: the first contains 4th and higher order terms, the second group starts with second order terms, and the last group is purely linear. Therefore, S cannot influence the linear part of (5.100). Also, in the region 134 of interest to us the second and third order terms dominate over fourth and higher order terms. In fact, there is a good chance that in the region where fourth order terms dominate, the generating function is not even defined anymore, especially since we are working with Taylor expansions. Hence, for best results, we have to choose S in such a way to minimize the second and third order terms in (5.100). Clearly, this is achieved for any M by s = 0. (5.101) With this result, it can be concluded that the optimal symplectification is achieved by the class of generators [S] obeying S = 0, and associated with the following a: — J M ‘1 J aopt : ( %M_1 %I > - (5.102) Interestingly enough, it turns out that if in aop, the linear part M is replaced with the unit matrix I, the resulting matrix gives a valid generator type, which exists for symplectic maps close enough to identity. It was first used by Poincaré in the restricted three body problem for a completely different purpose [39], and hence called the Poincaré generating function. Our aop, can be regarded as a dynamically adjusted Poincaré generator, to symplectic maps not having identity as linear parts. That is why we call it the EXtended POincaré (EXPO for short) generating function type. Finally, the best symplectified map .M,,,,, in the sense presented in this chapter, is obtained if the symplectification is performed using the EXPO generator type, i.e. Nopt E NEXPO- (5.103) Many examples of EXPO can be found in chapter 4. Indeed, the pictures confirm the conclusions of the optimal symplectification theory, i.e. EXPO is in general by far superior to any other generator type studied, including the traditionally used 135 conventional Goldstein types. It is expected that the EXPO will become the method of choice for symplectic tracking in COSY INFINITY. 5.5 Some Further Remarks In general, Hofer’s metric is rather little understood. It determines a kind of C"1 topology on HamC (Rzn), and gives rise to a paradox. Sikorav [69] showed that every one-parameter subgroup of Hamc (Rh) remains a bounded distance from the identity. So, we could draw the paradoxical conclusion that, according to the theorem proved in subsection 5.3.4, every point has a flat neighborhood, but in some sense a positive curvature is apparent. In any case, geometry on the group of Hamiltonian symplec- tomorphisms with respect to Hofer’s metric gives rise to a different way of thinking about Hamiltonian dynamics, and we can expect quite some progress in this direction in the near future. The understanding of the global features and properties of Hofer’s metric could give insight into the long term properties of Hamiltonian systems, and perhaps provide an exciting method to compute the region of stability without the time intensive tracking. Finally, we mention that there exists another, related metric on Hamc (Rzn), in- troduced in [66]. However, in [63] and [65] it was proved that they coincide locally on R2", so we need not consider it. We use Hofer’s metric because it is easier to work with. Put in a different perspective, according to (5.88), since we are working with Taylor expansions of generating functions, the best generating function is which has the fastest convergence inside its support. The first thought to minimize the left hand side is to choose the conformality factor ,u as large as possible. However, this is deceiving as has been seen in section 3.2. Here, another method is presented to the 136 same effect. Consider any admissible a, of the form (5.32) with ,u = 1. Compute the corresponding distance from (5.88). For another a in the same class, which differs just by )1 >> 1, the corresponding generating function is equivalent to the generating function of a with u = 1, subjected to the transformation a, I——+ pal. However, we have seen in the section on transformation properties of generating functions that this transformation is equivalent to F )——> pF. Hence, we get the same distance in both cases. At the same time, consider the transformation a2 1—2 #02. It follows from (3.27) that the generating functions transform according to F r——-> ,uF o u‘ll'. The simultaneous transformations a, »—-> pa, and a2 +——-> ,a‘laz leave the conformality factor equal to 1 and give F r—> F 0 ul, according to (3.21). Hence, inserting them into (5.88), again we get the same distances. It follows that the conformality factor does not add any flexibility. Therefore, the conformality factor does not influence the outcome of the symplectification, hence u = 1 is always assumed. As a side note, we mention that the conformality factor does play a role in the numerical stability of the algorithms. For example, if a generating function with [I S ||>> 1 is computed, (some entries in) the primitive function might be scaled down to zero on a computer (which in reality are tiny, but still non-zero), and it might be obtained that the generating function vanishes, 'which is obviously wrong. This follows from the fact that I/2 + JS/2 z JS/2 if I] S ||>> 1, and if we write S on 2111, then u can be considered as a conformality factor induced by the transformation a2 r—-> nag, which in turn induces the F v———-> pF 0 [1,—II transformation of the primitive function. Hence, if u > 1 numerical problems arise in the nonlinear part of F. This constitutes another reason for working with u = 1 only. Intuitively, Hofer’s metric measures what is the minimal Hamiltonian needed, averaged over time, to generate a symplectic map from identity. Actually, Hofer himself calls the distance from the identity of a map the symplectic energy of the 137 map. In view of the KAM theorem for symplectic integration methods [70, 2], which roughly says that the behavior of symplectic maps obtained by symplectic integration applied to a integrable system should be close to the integrable system (at least in the region where invariant tori still exist), this provides a strong support for our closeness criterion for optimal symplectification based on Hofer’s metric, and points out that Hofer’s metric is a natural choice. There is an alternative interpretation of Hofer’s metric applied to beam dynamics. Notice that, due to bi-invariance of Hofer’s metric we have p(M,N) =p(N-16M,I). (5.104) Therefore, by computing the distance between symplectic maps in Hofer’s metric, actually what is computed is the minimum Hamiltonian needed to generate the initial conditions of N from the initial conditions of M, such that both M and N applied to the respective initial conditions give the same final results. If the result of the norm minimization is small, the two sets of initial conditions are connected by a map close to identity, hence the two sets are close to each other. From a practical point of view this can be considered as not knowing exactly the properties of the injected beam. Such small uncertainties always occur in practice, and to be able to build robust accelerators, one should look for realizations that are not too sensitive to small errors. This is done by considering different error sources, each realization of the errors giving a different symplectic map. The incoming beam quality can be considered one type of such an error effect. Hence, from this viewpoint it does not matter whether M or N is used for the simulation of the accelerator. The optimality condition can be given a nice geometric interpretation. Minimizing the norm of the generating function is equivalent with finding the generating function type, which maps the Lagrangian submanifold determined by the symplectic map as 138 close as possible to the zero section. In fact, for any symplectic map, a generating function type can be found, such that it is globally defined and it is identically zero. This corresponds to the case when the corresponding generator type maps the Lagrangian submanifold exactly to the zero section. However, in practice this obser- vation is worthless, because the problem of finding the respective type of generator is equivalent with representing the symplectic map on a computer, rendering the symplectification procedure pointless. To see this, just replace in (5.89) M with the total map M. Nevertheless, the observation it is very useful in another sense. In the present chapter the optimality condition has been formulated for the most general case, but the solution has been found only for generator types of linear a, which was enough for the symplectic maps of practical interest, i.e. weakly nonlinear. In case M is strongly nonlinear, nonlinear aa are needed to map the respective Lagrangian submanifolds closer to the zero section. Thus, nonlinear a type of generators become a necessity, since with nonlinear as the Lagrangian submanifolds can be mapped closer to the zero section than with any linear a generator type. This observation hints towards the right direction to look for the appropriate nonlinear as. There are also other interesting consequences of these results, specifically concern- ing symplectic integration. The only difference between symplectic integration and symplectic tracking is the method for obtaining the truncated generating function. In the case of symplectic tracking, it is obtained from M,, using (2.9). In symplec- tic integration, it is obtained by direct solution of the generalized Hamilton-Jacobi equation (5.69). Thus, in this case, the linear part M is not known. Therefore, the prescription for Optimal symplectic integration is to use Poincaré’s generating func- tion and a sufliciently small time step (by this keeping the linear part of the resulting symplectic map as close as possible to identity). Moreover, if the Poincaré generator is expanded in a power series in the time step, from the Hamilton-Jacobi equation it 139 follows that, as first order approximation in time F p (w) = H (z). The corresponding integration method then reduces to the well-known implicit midpoint rule. To conclude this chapter, we would like to make a clearer connection between Hofer’s metric and the visual aspects of the assessment of optimality of symplectifica- tion. That is, while the optimal symplectiflcation theory says that in general EXPO is optimal, it has been seen that, in the case of the anharmonic oscillator of section 4.2, the generator type 5,, gave visually (or equivalently, in a 0" sense) better results than EXPO. What is the explanation in the light of the optimal symplectification theory? Well, S, is “more optimal” than EXPO because over the region which in- cludes all trajectories of interest, the Hofer distance between the exact map and the symplectified map with S, is smaller than the distance between the exact map and the map symplectified with EXPO. Unfortunately, we cannot compute Hofer’s dis- tance exactly. However, considering the 19th order Taylor map as the exact solution, according to (5.86) we can approximate the distance very accurately from dEXPO z“ FgXPO — FEXPO H, (5.105) and d5, 81]] F159" — F15,” H, (5.106) respectively. We plotted F19 — F11 (magnified by some constant factor) with Mathe- matica for the two cases, and we obtained Figure 5.1. We can read off the plots dEXPO z 3, (5.107) Obviously, S, is better than EXPO. Notice that the sharp deviation from zero of the distance function occurs only in the region close to the dynamic aperture, where we already noticed most of the discrepancies among the various maps. This observation I40 (a) EXPO symplectification (b) S, symplectification Figure 5.1: Plot of Hofer’s distance between the exact and the symplectified maps over the region enclosing the dynamic aperture of the anharmonic oscillator studied in section 4.2. 141 opens the prospect of using Hofer’s metric as a local indicator beside the global one that we already know. 142 Chapter 6 Summary of Part I The extended theory of generating functions of canonical transformations was de— veloped. Using a modified definition of the generating function, it was showed that there are many more generating functions than commonly known, and some always are defined globally. The set of generating functions turned out to be very degenerate from the symplectification point of view. However, employing some transformation properties of the generating functions, it was possible to reduce the pool of generat- ing functions to equivalence classes associated with linear conformal symplectic maps. The remaining independent types were characterized by symmetric matrices. Also, it was proved that by choosing apprOpriate types of generators, there is no advantage in factoring out linear parts and symplectifying only nonlinear parts, or first subjecting the map to be symplectified to a linear symplectic coordinate change. The performance of this symplectification method was illustrated by a variety of examples. It was showed that different generator types often give significantly different long term behavior of the symplectified maps. This fact pointed out the necessity for optimal generating function symplectification studies, which were solved in a very general setting based on Hofer’s metric. Using Hofer’s metric, a condition for optimal symplectification was given. After 143 a few manipulations, Hofer’s metric for Hamiltonian symplectic maps was expressed in terms of associated generating functions. Therefore, finding the best symplectified map turned out to be equivalent to finding the appropriate generator type. It was shown that the generator type which satisfies the optimality condition in general is given by (5.102), and it was called the EXPO type. Consequently, the symplectic map N E X p0, obtained by symplectification of M,, using EXPO, gives optimal results in general. However, these results do not exclude the existence of custom tailored generator types that give better results for specific symplectic maps. EXPO is not only optimal, but it is also fast due to its simplicity, implementation using fixed point iterations, and vectorization. Throughout part I methods of symplectic geometry and topology have been uti- lized. The unifying concept behind these methods is the flows of Hamiltonian systems. The accelerators of interest to us can be modeled as Hamiltonian systems. The time evolution of these systems (i.e. the flow, or the particle orbits) can be regarded as curves on the space of Hamiltonian symplectic maps. The geometric properties of these curves with respect to Hofer’s metric are deeply related to the dynamics. In this dissertation we exposed and exploited several aspects of this relationship. More precisely, the Hamiltonian system representing the real accelerator lattice is replaced by another Hamiltonian system, and instead of simulating approximately the real accelerator, we track exactly a nearby accelerator, where closeness is measured in Hofer’s metric. Then we hope that the perturbations so introduced are small enough that in the light of the KAM theorem most invariant tori, and hence most of the geometric structures, survive, and it leads to a more reliable and faster estimation of the dynamic aperture. Finally, while the results obtained in this dissertation have been derived with accelerator physics motivation in mind, their relevance go beyond beam physics, and 144 directly apply to any other weakly nonlinear problem in Hamiltonian dynamics. 145 Part II Accurate Simulation of Eringe Field Effects 146 Chapter 7 Introduction and Theoretical Background In part I it was shown that symplectification of the truncated Taylor maps, in general, is necessary, and it must be done by using the appropriate generating function type, EXPO. As a prerequisite of symplectification, it was assumed that the truncated, order n symplectic, Taylor maps represent the systems well over short times. If this is not the case, it is easy to find examples for which no symplectification can give good results. For example, consider two truncated Taylor maps representing two different systems, which agree through order 17., but differ by a large amount starting at order n + 1. This can happen, for example, if the amplitude dependent tune shifts of both systems agree up to order n, but only one of the system has some very large tune shifts at order n + 1. This entails that no symplectification can predict the behavior of the system with large tune shifts, if the order n truncated Taylor map is used for symplectification. The reason is that important information was neglected about the system, which renders the order n truncated Taylor map inaccurate, even for short times. Therefore, for successful symplectic tracking, the order n needs to be high enough (the exact value being system dependent), and every relevant effect must be incorporated into the Taylor map. One of the effects that requires more sophisticated 147 methods for their inclusion in'the one-turn Taylor map, are the effects of fringe fields. In part II of this dissertation, the theory relevant for accurate fringe field effects simulation is developed, the effects are studied generically in chapter 8, new methods for multipole decomposition of superconducting magnets are introduced in chapter 9, and the results are applied to a few examples. For the LHC, a detailed study draws some conclusions about the effects of realistic fringe fields on the nonlinear dynamics of particles in chapter 10. Due to the diversity of the field of nonlinear beam optics, the mathematical meth- ods employed and the formalisms utilized can be very different depending on the specific design requirements. One of the topics for which traditionally very different approaches have been carried out in different subfields are the fringe fields, or end fields. For the purpose of simulations of large storage rings, fringe field effects are often neglected. Sometimes this is a quite good approximation. However, strictly speaking, it is an unphysical model, as the electromagnetic fields of the model do not satisfy Maxwell’s equations. The simplest method to take fringe fields into ac- count is to approximate their effect with a kick (i.e. impulsive momentum change, while position is unchanged) characterized by the integrated field value [71]. While this model may alleviate some problems, it is not a cure, and more sophisticated models are needed for accurate simulations. Besides the kick, the effect of the fringe fields has been characterized by a sudden shift in position at the so-called effective field boundary [72, 73, 74, 75]. For a specific field falloff, in [76] the third order aberrations and their scaling with fringe field extension have been computed for the quadrupole. Also, leading order hard edge fringe field effects have been studied in [77]. Fringe fields have been shown to adversely affect the PEP-II dynamic aperture [78]. However, an extensive study of fringe field effects has so far not been available. The nonlinearities due to fringe fields have been well-known in the field of high 148 resolution particle spectrographs for a long time [74, 79]. Also, as recently has been shown, they tend to become significant in small rings, especially at larger emittances [11, 12]. The latter studies motivated us to look more deeply, at a fundamental level, at the fringe field effects. The introduction of the methods used to study the fringe field effects, first requires the development of a theoretical framework. In the following we assume straight optical axis, and a source-free region surrounding the axis. In this divergence-free, curl-free region of the magnets it is possible to derive the magnetic field components from a magnetic scalar potential that satisfies the Laplace equation. The general solution expanded in cylindrical coordinates with axial coordinate s, in the so—called multipole expansion form is [80] oo ’5 = Z (bk,,(s) sin to + ak,,(s) coslo) rk. (7.1) 21:0 The functions b,,,(s) are called the normal and a,,,(s) the skew multipoles, respec- tively. The components, according to l = 0, 1, 2, 3, are called the solenoid, dipole, quadrupole, sextupole, etc. components, respectively. Defining 9k,,(s) and M,,,(s) by tan6k,,(s) = —:::((:)), (7.2) M,,(s) = (/bz,.+at,., (7.3) we have an equivalent form VB = Z M,,,(s) cos (1.) + e,,,(s)) rk, (7.4) k.l=0 which Shows that any normal (skew) component can be obtained from the corre- sponding skew (normal) component by an s-dependent rotation around the 8 axis. 149 The link between the two forms in the other direction is bk'((8) = —lhfk‘1(8)Sln6k‘1(8) (7.5) (lvk,((8) = A’Ik‘z(S)COSQk‘1(S). Inserting (7.1) in the Laplace equation in cylindrical coordinates _ 1 0 761/3 1 82V3 82VB _ AVE—767(ar)+73537+—652_‘0’ (76) we obtain 2 (WASH? - 12) + ”if—2.48)) Sin “.5 + (alc.z(8)(k2 —12)+ 025—210)) 608105] T” = 0. lc,l=0 (7.7) 7 using the convention that the coefficients vanish for negative indices. Due to the fact that the above equation must hold for every r and ,5, and the sin and cos are linearly independent, it follows that bk.z(8)(k2—12)+ [5-2.1(8) = 0 (7-8) ak,,(s)(k2 —12) + az_2,,(s) = 0. (7.9) Furthermore, it can be shown that the following recurrence relations hold: b (s) = blinks) (710) ”2"” 113:, <12 — (1+ 2102) ' (2n) _ “1,1 (3) and the coefficients that cannot be obtained by these relations are zero. The terms that contain s-derivatives are called pseudo—multipoles. It is worth mentioning that, as one can see from (7.2), the recurrence relations (7.10) do not hold in general for 150 M,,,(s). However, when the Bus are s-independent, (7.10) holds for M,,,(s). Inserting relations (7.10) in (7.1), we get for the potential VB 2 Z (f,(r, s) sin lgb + g,(r, 3) cos lob) r’, (7.12) (:0 where (lb(2n) (8 ) 2n flfflsl :: Elm-211,“)! =2 anl —(l+ 2V)2 )7‘ (7.13) 6,226) 2 bits) ._ (its) . ‘ (“(8) — 4(1+1)r + 32(1+1)(1+ 2) 384(1 + no + 2)(1+3)T + " oo (2n) a (8) =ZGl+2nl(S)’ =12; II"=1 2,111+ 2V), 2)r2" (7.14) at’e) 2 alts) , alts) = 011(3) - 404—1)" + 32(z+1)(1+2)f _ 384(l+1)(l+2)(l+3) r6+... The functions f,(r,s) and g,(r, s) represent the “out of axis” expansion of the mul- tipoles. The magnetic field components in cylindrical coordinates can be calculated using the well known formulas resulting the expressions B, (r, (b, s) = B¢(T‘, ¢, 8) : Bs(r, a, s) = _ avg B, — 85 (7.15) _ 1 VB B, _ T 89 (7.16) _ av), B, _ E’ (7.17) g0(r, s) + Z [fi(r, s) sin lgb + g,(r, 3) cos lob] r"1 (7.18) l=1 Z[l( f,( (r, s) cos lo g,(r, 5) S111 l¢)] (7.19) (:1 Z [f,'(r, s) sinlcp + g](r, 3) cos to] r’, (7.20) (:0 151 where prime denotes derivative with respect to s and °° °° (l+2n )b]2n)s() 2.. (7‘ 8) : :L_:O( (Id—271) )b[+2n[(8 =nzzo 1111:1(“2 (1+ 2V)2 )7‘ (7.21) 14.25“” 5 1+4 5“” 1+6b§ff Zlbl,[(S)—( )l,l( ) 2 ( ) (3) 4 ( ) (8) T6+... 4(z+1) T +32(1+1)(1+2)’” —384(l+1)(l+2)(l+3) ~ 001+ 2n )alen) (s) 2" 910', s) = "231+ 2n)a1+2n,(81=z H3: (12 _ (1+ 2”) )r (7.22) _, ()_ (1+2)a§.’s( 8),, (1+4)a§‘l(s) T,_ (l+6)a§f7(s) T6+ _ a” 8 4(l+ 1) 32(l+1)(l+2) 384(l+1)(l+2)(l+3) It can be seen that every multipole, except for l = 0, is multiplied by rl‘l. For the special case I = 0, we obtain B14738) = ="—Z(1)k+lmah2ri)(3)r2k_l (7-23) k=1 B¢,(r,s) = 0 (7.24) 3,035) = w:1)k+122k:!k!532§+1>(s)7~2k. (7.25) k=0 In the Differential Algebraic (DA) picture, the field calculations are done locally, as Taylor expansions of the fields with respect to the Cartesian coordinates x, y, 3. Hence, we need the following equations relating the cylindrical and Cartesian components of the magnetic fields: Bx(r, o, s) = B,(r, d), s) cos 05 — B¢(r, (p, s) sin 05 (7.26) By(r, 05, s) = B,(r, a, s) sin (b + B,(r, gt, 3) cos d), (7.27) and Bs(r, ab, 3) is unchanged. 152 Chapter 8 Generic Efiects For a better understanding of the fringe field effects, we start their study in this chapter, by performing a study of the effects that one may miss by not considering the influence of the fringe fields. We keep the study generic, and use lattices of the muon storage rings, which, however, are available only at a preliminary design stage, and to illustrate general trends in small footprint (i.e. small effective area occupied by the accelerator), large acceptance rings. Specifically, we use a version of the proposed 30 GeV Neutrino Factory [52], and of the FNAL Proton Driver [53]. The fringe fields’ falloff are modeled by a six parameter Enge function F(s) = 1 (8.1) 1 + exp (2 a.- (s/DY“) i=1 where s is the arclength along the reference trajectory, used as the independent vari- able. D denotes the full aperture of the magnet, and the a,- (i = 1, 2, ..., 6) are called Enge coefficients. For practical calculations, the Enge coefficients are either fitted to represent measured data [81], or obtained by multipole decomposition of detailed field computations as in chapter 9. We look at fringe field effects as a function of magnet aperture and falloff shape. This is achieved by varying the magnet apertures D, and the Enge coefficients (1,. 153 The fringe field effects can be particularly easily studied in the map picture using Differential Algebraic methods [82, 9, 10]. The consequences of the fringe field effects influence all orders of the motion, beginning with the linear behavior. The pseudo- multipole nonlinearities of the fringe fields couple to higher derivatives of the multipole strengths. In practice this entails that fringe field effects become more and more relevant the more the particles are away from the axis of the elements, which, of course, is directly connected to the emittance of the beam. Also, it is clear that the falloff shapes, and implicitly the apertures, have an influence on the induced nonlinearities. Complete treatment, to any order, of detailed fringe field effects is possible in the code COSY INFINITY [83, 84, 24, 85]. To quantify the effects, we compute linear tunes, amplitude dependent tune shifts, Chromaticities, resonance strengths, and estimate dynamic apertures. In the following sections we present observations related to aperture (section 8.1) and shape dependent effects (sections 8.2 and 8.3). In section 8.4 it is presented briefly that neglect of the high order pseudo-multipoles can also give misleading simulation results. Moreover, in section 8.5 it is pointed out that, in general, symplectification is not sufficient to undo the errors made in neglecting important information about the system (such as higher order pseudo-multipoles). 8.1 Aperture Dependent Effects on Linear Tunes and Chromaticities In some perturbation theories the linear or first order effects are not considered. In others, the first order effect is characterized by a kick [71], or as a sudden change in position and momentum at the so—called effective field boundary [74]. Here, using the 30 GeV Neutrino Factory lattice, we compute with COSY INFINITY the linear tunes and Chromaticities as a function of the magnet apertures. We assume that all 154 Dipole Quadrupole Sextupole al 0.478959 0.296471 0.176659 a2 1.911289 4.533219 7.153079 a3 —1.185953 —2.270982 —3.113116 a4 1.630554 1.068627 3.444311 a5 —1.082657 ——0.036391 —1.976740 a6 0.318111 0.022261 0.540068 Table 8.1: Enge coefficients of the default fringe field falloff used in COSY INFINITY for dipoles, quadrupoles, and sextupoles, respectively. the magnets have the same aperture, and the falloff is given by the Enge coefficients of Table 8.1 [86]. We will call them hereafter the default Enge coefficients. It is important to note that the Enge function model can be used for a global fit of the magnetic fields, including the out of axis expansion. This has been demonstrated in several real situations, as for example the NSCL’S S800 spectrograph [87], the G81 QD kaon spectrometer [88], and even the rather peculiar LHC HGQ lead end [81]. The aperture is varied between 1 mm and 300 mm. Fig. 8.1 gives the results for the :1: and y center tunes (tunes in the linear approximation). In the stable regions the tunes change continuously and monotonically with the aperture. However, in general there is a nonlinear relationship between center tunes and the aperture. As the linear motion is uncoupled, for linear stability the absolute values of the traces of the :1: - a (horizontal phase plane) and y — b (vertical phase plane) submatrices respectively need to be less then 2. The variables a, b are scaled momenta, conjugate to a: and y, respectively. The nonlinear dependence of the traces on the apertures is also clear from Fig. 8.2. The trend regarding Chromaticities is included in Fig. 8.3 and 8.4. It has been noticed that the main impact of the fringe fields is coming from only a few matching quadrupoles in the arcs [12]. We repeated the computations of this section for the same ring, with the respective matching quadrupole fringe fields turned 155 03 xx 0.8 x x OJ +*+i 05 [+ I+ 05 Center Tune 0.4 + + 0.3 ,. + XX 0.2 xx) + 0.1 YX + . t i 4 h 0 50 100 150 200 250 300 Aperture (mm) Figure 8.1: Center tunes as a function of aperture. The fringe field shape is given by the default Enge coefficients. off. As one can see from Fig. 8.5 and 8.6, the results are different only quantitatively, but qualitatively the situation remains the same. 8.2 The Sharp Cutoff Limit As already mentioned, the sharp cutoff, or hard-edge, approximation is a contradiction in itself, as far the physics goes. However, as a purely mathematical approximation, it still can be analyzed in some detail. Qualitatively, it can be characterized as follows. The function that describes the falloff is called the cutoff function, or a bump function. It is well-known in the mathematical literature [43, 89] that infinitely often continuously differentiable (C'°° smooth) bump functions can be found such that they take the value 1 on one closed set, and assume the value zero on the complement of another closed set; one of the closed sets lying in the interior of the other closed 156 3 r xtrace + ytrace x yXXxxxXxX x X "’ xx x + x l‘ x xx X + 1 Xx x 4- )Kxx X xx x + Xx X + x + x + 0 X 1+ ° mm... 4 o E +++ + )— +++ + 3 ++ + ++ + x +[ + ++ + X ++ ++ 2 +“‘+ ‘r X ++++++++++ X x x -3 X X [r -4 O 50 100 150 200 250 300 Aperture (mm) Figure 8.2: Traces of the a: - a and y — b submatrices versus aperture. The fringe field shape is given by the default Enge coefficients. set. For our case the inner closed set can be taken as the region of the main field of the magnet, and the complement of the outer closed set the region where the fields practically vanish. Furthermore, the two sets are arbitrary except the already mentioned conditions. This means that the two sets can be taken arbitrarily close to each other in some sense (for example in the Hausdorff metric). Thus, the falloff speed can be arbitrarily fast, and at the same time the smoothness can be kept intact. This is why the sharp cutoff limit gives satisfactory numerical results when fringe fields are not important: in principle, the falloff regions’ width always can be taken smaller than the smallest step of any integrator, and thus escapes from any numerical issues. On the other hand, as a rigorous mathematical limit, the sharp cutoff limit corresponds to the case where the two closed sets “touch” each other. In this case, any bump function tends to the sum of two Heaviside functions, which then end up in 157 :8] >1 3500 2500 x 2000 Chromticities 1500 1000 xx Xxxx xx ){XXXX 0 .1..I__1..;_ LLLLLLLLLLLLLL W+ALLL .L_L.L.A_AL+JT_+_L_ ALLAW AAXX XXXMXXX**){§;TTT Trf' ryrrv v Ivvv X X 250 300 Aperture (mm) Figure 8.3: First and second order :0 Chromaticities as a function of aperture. The fringe field shape is given by the default Enge coefficients. the Hamiltonian. However, as it is well-known, the dynamics is not governed by the Hamiltonian itself, but by its gradient. The Heaviside function’s derivative being the Dirac delta function, it follows that the sharp cutoff limit gives rise to divergences. The divergences show up as blowing up of some of the map elements, and, as a consequence, some of the tune shifts and resonance strengths. Hence, blow-up of amplitude dependent tune shifts in the sharp cutoff limit occurs regardless of the exact shape of the falloff. However, in the perturbative order-by- order approach, the divergence can occur at different orders depending on the specific fields involved. For the case of the homogeneous dipole, it has been derived [90] that the lowest order map element that causes divergence is (blyyy), that is the element that shows how the final angle in the vertical phase plane depends on the initial position in the vertical direction. 158 X X XxXxxXx 0 LLLALLAL AAAAAAAAAAAAAA an mmmmmmmm «h ALLA AAAAA ALAL+ §§;;)TTTTT..TT ++TTY .. ----xxmr -.--4H . -- W (X xx Xx X -200 Chromaticities 8 o 50 100 150 200 250 300 Aperture (mm) Figure 8.4: First and second order y Chromaticities as a function of aperture. The fringe field shape is given by the default Enge coefficients. For multipoles with straight optical axis, the divergences occur at higher orders. Recently, under some simplifying assumptions, estimates for the second order ampli- tude dependent tune shifts of a quadrupole fringe field have been calculated [91]. It is shown that, in the integrated field approximation, the second order tune shifts tend to a finite non-zero value in the sharp cutoff approximation. This is derived from a few integrals that are part of the integrated Hamiltonian, and taking limits as the extension of the fringe region goes to zero. Within its domain of validity, the estimate gives good agreement with the exact values computed by COSY INFINITY, at least for the cases studied. The second order tune shifts (i.e. the quantities that represent the quadratic dependence of the tunes on the amplitudes) are functions of the third order map elements. To compute the third order map elements of a quadrupole, integrals of the 159 xmne + xxxxXx ymne X XXX 0.9 0.8 Im++++ A“ 0.7 "V'T++++++ «[v+ ++++++ ++ 0.6 ‘* + + ‘r+ ++ ++ + 0.5 a. Center Tune kx XXxxxxx Xx Xxx Xxxxxx J 0 50 1 00 1 50 200 250 300 Aperture (mm) I x trace + [I trace x 2 xxxxxxxx“""fiixxxXXXxxxx xxxxJ Trace -1 ++ _2 ‘f++4.+++1 ++++++++ AA 'TT -4 so 100 150 200 250 300 Aperture (mm) Figure 8.5: Same as Fig. 8.1 (top), and Fig. 8.2 (bottom), but without the matching section, where most of the fringe field effects are concentrated. 160 20 x1) + x2) x + + + ++++"“t'++-l- 04W++++ +++++++++ ++g¢++++++ ++++++¢++ +++ X71 + Xx + xx x ~20 "x x m x a: x ...: x '6 x .3 x q; X E -40 x 2 .c: x o x x x -60 x x x x x -80 “ x x -100 50 100 150 200 250 300 Aperture(mm) W) + y(2) x XXX xxxXXxxxXxX1k xxxx XXXXXXXX xx _20 xxxxxxxxxxx 0) .93 .2: .9 on E -40 9 .c xx x x -60 x x x -80 -100 50 100 150 200 250 300 Aperture (mm) Figure 8.6: Same as Fig. 8.3 (top), and Fig. 8.4 (bottom), but without the matching section, where most of the fringe field effects are concentrated. 161 type 3e 12 = / r3 (3)5212 (s)ds (8.2) 8b appear, where 52,2 (3) is the second order derivative with respect to s of the quadrupole strength, and 3b and 38 are the beginning and end of the fringe field, respectively. The function r (s) is the initially unknown orbit. To evaluate the above integral within the framework of perturbation theory, one can distinguish two cases. In the approxi- mation that the orbit does not change over the fringe region, that is r (s) =constant over 3 6 [3b, 38], the result is 12 = "r3 (85) (52,2 (36) —— 52,2 (53)) . Because at the begin- ning and end, the fringe fields have assumed their plateaus b(s) = 0 and b(s) = 1, respectively, we have 52.2 (36) = bin (3),) = 0, and hence altogether 12 = 0. This approximation yields the prediction that there are no second order tune shifts, inde- pendent of the specific shape of b2,2(s). On the other hand, we obtain a different, and more precise, answer performing perturbation theory successively over small equidistant subintervals of the interval [5b, 88], which in the limit of all subintervals going to zero leads to the correct result in much the same way as numerical integration schemes. Again we assume pertur- batively that r(s) is constant over the interval in question, which in the i-th step of the perturbation theory spans 3 E [3,, s,+1]. If A3 = 3,+1 — s,- is small enough, this approximation will become better and better. However, unavoidably r(s) will change slightly over different time steps. Consider the specific case of performing N sub-steps 162 of the perturbation theory. Then an estimate for [2 is N 3i+1 N 12%er (3.) / b§,.(b;,2 12 z — ZCH-l Z lej (7'3) J (8i) . (8.11) 163 In the sharp cutoff approximation, D —> 0, only the j = 1 terms survive the limit. Thus, finally we obtain the qualitative behavior N I — ZcI+1-(r3) (3.) (8.12) i=1 where we absorbed various constants into c,‘ +1. Instead of being zero, the integral now approaches a constant and usually nonzero value as D —> 0. Therefore, the convergence to a constant of the second order quadrupole tune shifts in the sharp cutoff limit can be qualitatively understood. We mention that the limits of the extension of the fringe fields or the size of the apertures going to zero are equivalent. On the other hand, in the case of fourth order tune shifts in the sharp cutoff limit we obtain completely different qualitative behavior. The fourth order tune shifts are functions of fifth order map elements. In this case we need to estimate integrals of the type 3e 14 =/ 1"”r’(. 7r and I71] 3 Ill; but this corresponds to a point in the close proximity of the wire. The necessary ingredients for the DA field calculation are the above formula and the analytic model of the magnet, consisting of line wire currents. To this end, the entire field in space is calculated by summing up the fields created by wire currents. At each step, the evaluation of (9.11) in DA yields not only the value of the magnetic field at the respective point, but also its derivatives, that is the Taylor expansion with 188 Be (13,841) By (in, s, y) Coefficient Order Exp. (33,3,y) Coefficient Order Exp. (x,s,y) 0.93844739E—11 0 0 0 0 -.83395100E-11 0 0 0 0 0.10229458E—09 1 1 0 0 17.49010593690 1 1 0 0 0.14301198E—09 1 0 1 0 -.13045929E—09 1 0 1 0 17.49010593689 1 0 0 1 0.95844482E-10 1 0 0 1 -.60674194E—09 2 2 0 0 0.55438195E-09 2 2 0 0 0.36025250E—08 2 1 1 0 -3.80475173597 2 1 1 0 0.12053635E—08 2 0 2 0 -.11068638E-08 2 0 2 0 -.20145723E—08 2 1 0 1 0.19416855E-08 2 1 0 1 -3.80475173613 2 0 1 1 0.34442337E—08 2 0 1 1 -.59862145E—09 2 O 0 2 0.55248205E-09 2 0 0 2 -.22827223E—07 3 3 0 0 -1.66625730542 3 3 0 0 -.92745885E—09 3 2 1 0 0.12449044E—09 3 2 1 0 0.71839210E—07 3 1 2 O 9.99754389589 3 1 2 0 0.55577595E—09 3 0 3 0 -.21654983E-10 3 0 3 0 -4.99877197923 3 2 0 1 -.30524801E—08 3 2 0 1 -.11331539E-06 3 1 1 1 0.11068027E—06 3 1 1 1 9.99754389364 3 0 2 1 0.69547317E—07 3 0 2 1 -.33575741E—08 3 1 0 2 -4.99877197953 3 1 0 2 -.739876993E-09 3 0 1 2 -.59531324E—10 3 0 1 2 -1.66625730477 3 0 0 3 -.22164947E-07 3 0 0 3 Table 9.3: Taylor expansion of the magnetic field components Bx and By. The columns represent the expansion coefficients, the order in the expansion and the exponents of (:r, 3, y), respectively. respect to coordinates. The result for the return end of the Large Hadron Collider’s High Gradient Quadrupole, up to order 3, is presented in Table 9.3. The correctness of our results have been checked against data obtained from Fermilab. The Fermilab data contains the values of the components of the field on the surface of a coaxial cylinder with the Optical axis, and have been supplied by G. Sabbi. To Show that the Biot-Savart law implementation based on (9.11) is much more stable then, for example, based on (9.6), we use as indicator the 3 component of the curl of the field. From Table 9.4 it is clear that the naive implementation goes wrong as low as second order in the curl. Due to lack of space, we presented the result only 189 (V x I?) from (9.11) (Y x E) from (9.6) Coefficient SOrder Exp. (x,s,y) Coefficient Order Exp. (x,s,y) 0.64481753E-11 0 0 0 0 0.64517280E—11 0 0 0 0 0.31233362E-08 1 1 0 0 0.31233356E—08 1 1 0 0 0.15774093E-09 1 0 1 0 0.15770096E—09 1 0 1 0 0.31389284E—08 1 0 0 1 0.31389279E-08 1 0 0 1 0.62971751E-07 2 2 0 0 0.336831253164 2 2 0 0 0.11356437E—06 2 1 1 0 0.11356393E—06 2 1 1 0 0.22546586E—08 2 0 2 0 0.22374493E—08 2 0 2 0 0.61018801E—09 2 1 0 1 0.61014426E-09 2 1 0 1 0.11216003E—06 2 O 1 1 0.11215921E—06 2 0 1 1 -.65226274E—07 2 0 0 2 -.336831255411 2 0 0 2 —.10351639E-06 3 3 0 0 0.84829867E-06 3 3 0 0 0.36038492E-05 3 2 1 0 -1010523.36223 3 2 1 0 0.23104571E—05 3 1 2 0 0.44302147E-06 3 1 2 0 0.23642363E-07 3 0 3 0 -.31176547E—04 3 0 3 0 -.19778205E—05 3 2 0 1 -.38402254E—05 3 2 0 1 0.94348351E-08 3 1 1 1 0.94491708E—08 3 1 1 1 0.22561792E-05 3 0 2 1 0.62955066E-O5 3 0 2 1 -.199990078E-05 3 1 0 2 -.20901225E—05 3 1 0 2 -.367474004E—05 3 0 1 2 101052336227 3 0 1 2 -.927884933E—07 3 0 0 3 0.50706360E-05 3 0 0 3 Table 9.4: Comparison of the 3 component of the curl, up to order 3, computed by two different implementations of the Biot-Savart law ((9.11) and (9.6), respectively). up to order 3, but our results show that the behavior of (9.11) is good up to very high orders. Also, we mention that probably this is the most straightforward and accurate way to compute the curl and hence verify whether Maxwell’s equations are satisfied. 9.2 Multipole Extraction Algorithms 9.2.1 The Direct Method Using the field computation of the preceding section, it is possible to extract the multipole content of magnetic fields directly, in a very elegant way that is arbitrary in order. 190 Obviously, if we evaluate the above equations in the midplane (y = (15 = 0), then B,.('r, c5 = 0, s) In”: 31(1):, 3,; = 0, s), (9.12) B¢(7‘, cf) 2 0, s) |,,_,I= By(17, y = 0,3). (9.13) Therefore, all the information we need to extract the multipoles up to the order of calculation are the Cartesian components of the fields in the midplane B(.r,y=03)=go(:c,s) +Z;g(r (9.14) By(:r, y = 0, s) :2 21,002.33) -:z:"1. (9.15) (:1 This is possible due to the previously mentioned fact that any multipole strength of order I is multiplied by 27"]. Starting at l = 1, (11,1(3) is extracted as the :1:- independent part of B3, and analogously b1,1(s) from By. Evaluating (21,1(3) and b1,1(s) at s = 0 yields the skew and normal dipole components, respectively. From a1,1(s) and b1,1(s) the functions f1 (17,3), §1(.r,s) are generated up to the order of calculation and subtracted from Bx (a: r,—y -— 0, s), and By(.r,y : 0,3), respectively. This cancels the pseudo-multipoles generated by the s-dependence of a1,1(s) and b11(s) (see (7.10)), which otherwise would make the distinction between sextupole terms and pseudo-dipole terms impossible. The procedure can be iterated for the higher order multipoles, up to the order of calculation. After the k-th step, the remainder of the field components should contain just (k + 1)-th and higher order multipoles. However, there is an additional problem in the case of solenoidal fields (the case I = 0). In this case, we have an a.0,0(s) in the potential, but its contribution vanishes from the field components BI and By, so the function §o(.r, 3) cannot be generated from the information available in Bl. and By. Fortunately, it can be generated from 191 the B8 component, which evaluated at :1: = y =2 0 yields (28,0(3). From this function we can calculate (109(3) up to a constant and generate the l = 0 contribution to Bx, §0(:r, 3). Once this is subtracted from 3,, the method works as previously described, starting with l = 1. Finally, two notes: the method relies on the fact that the magnetic field can be generated by a magnetic scalar potential that satisfies the Laplace equation. There- fore, it is really important that the curl of the field vanishes. If the fields are calculated from line currents by the Biot-Savart law, that means that the model should consist only of closed circuits to ensure vanishing curl. Maxwellification of the field ensures a better numerical stability of the algorithm. Secondly, only in the regions where the magnetic field is not s-dependent the functions f; and g, are equal to the true multipoles, and [fl = fl, lg; = g,, an assumption that is sometimes made even for the s-dependent region too. 9.2.2 Multipole Extraction by Analytical Fourier Transform There is an alternate way to extract the multipoles. The field computation being performed in COSY in Cartesian coordinates (1:,y,s), it is possible to perform an analytical pseudo-Fourier transform, i.e. a series of coordinate transformation in DA, keeping throughout the r and s dependences. This is done by our S—Dependent Differential Algebraic Analytical Fourier Transform briefly presented below. Initially, the field components are in the form of (9.33). The first transformation is (:13, y, s) v——> (7“, cos 6), sin (15,3) by :r = 7‘ cos ()5 and y = r sin ¢. At the same time, using (9.35), we switch to cylindrical coordinates, obtaining the field components in the form of (9.39). Note that we are going from a 3 variable representation to a 4 variable one, hence some of the information in the new representation will turn out to be redundant for our purpose. The next transformation is (7', cos 4), sin qfi, s) +—-> (r, em, 6“”, s), that 192 is a complex exponential representation, using cosqb = (61¢ + e‘i‘f’) / 2 and Sing!) = (6” — er) / (22'). We show in section 9.4 that it does not matter which field component is used for Fourier transformation, so assume that we are working with B,. Then, we have it at this stage in the form B,(r, 6"", e‘w’, 3). Now, in principle, it is possible to recombine various products of powers of 6'7" and 6‘“ to trigonometric functions involving multiple angles. However, the key point is to notice that we can obtain the true multipoles by setting ew’ = 0. This is true due to the fact that all the terms of the form eiq¢e'i7’¢, with q, p # O, are responsible for the pseudo-multipoles. This becomes clear if one takes a closer look at (9.43). By setting eid’ to zero, we get rid of all the pseudo-multipole terms and we are going back to a 3 variable representation n+1 B,(r, e728, 3) = Z z,(s)e"“¢r’"1, (9.16) [:1 where 2:1(3) are complex functions of 5. By comparison with (9.57) it is obvious that one term of (9.16) can come only from [211(3) cos 1gb + Bl(s) sin lqb] 7"”1, which also can be expressed as [ea-““5 (552(3)- + egg—82) + e“¢5 (912(1) — 2412(2)] 7H. (9.17) After setting 6'” to zero and comparison with (9.16) we obtain the result Bz(s) = 2 Re (21(3)) and A,(s) 2 21m (21(3)). As a final step, we take the true multipoles as given by bu(8) ‘= — Re (21(5)), (9.18) az,z(s) = -l—Im(zl(s)). (9.19) 193 9.3 Enforcing Maxwell’s Equations 9.3.1 Local Maxwellification Given a magnet model consisting of line currents, it is possible to compute the mag- netic field generated by the Biot-Savart law in the DA framework as a local Taylor expansion with respect to the Cartesian coordinates (2:,y,s), as has been demon- strated in section 9.1. The magnetic field should be divergence and curl-free in the regions of interest, as implied by Maxwell’s equations in a source-free region: V-B = 0 and V x B = 0. This is the case only when the magnet model consist of closed loops of current. Realistic magnet models, as for example the LHC HGQ end regions as modeled by the code ROXIE and supplied by G. Sabbi of FNAL, are not closed due to presence of image currents, “leads” and separate treatment of the two end regions (lead end and return end). One way to fix this problem is to input as much physical intuition as possible to close the magnet model, compute the field generated by this model, which should differ as little as possible from the original model. This is the first step of Maxwellification. Obviously, the solution is not unique due to infinitely many ways of closing the model. The closing is important to guarantee vanishing curl, as it is required by the Maxwell equations, and in this case the field is deriv- able from a scalar potential. From the DA computational point of view it is also important, because it is enough to compute the field components only in the mid- plane (see section 9.2). The computer time needed for computing a magnet model of several 105 line currents to high order in one end region of the LHC HGQ’s scales much worse with the increase of the number of variables than with the increase of line currents, which should be linear. Besides computer time, the second step of the Maxwellification provides a way to correct for small computational errors or magnet model imperfections. One specific example is the method of the next section. A1- 194 though the curl is already small, as shown in Table 9.4, the numerical stability of the multipole extraction algorithm is improved by local Maxwellification of the field, which is described below. If we restrict ourselves to elements with straight optical axis for simplicity, the second step of Maxwellification proceeds as follows. Given Bx(;r,0,s), By(:r,0,s), Bs(a:, 0, s) (y = 0 representing the midplane) we can compute the field components in the whole space. From a scalar potential V(:1:, y, s) that satisfies the Laplace equation 62V(:r,y,s) 82V(a:,y,s) 82V(x,y,s) T+T+T—O’ (9.20) the field results from the well-known relation B (2:, y, s) = VV(:L‘, y, s) (we neglect the sign which is irrelevant in our discussion). We transform the Laplace equation to a fixed point problem by isolating the y derivative term and integrating with respect to y (92V(a:,y,s) 82V(:r,y,s) 82V(.r,y,s) T — ‘ (T + T) (921) y, 32V(I,y”,8) ,, 3” 32V(x,y”,8) 32V(~Tay”,8) ” /0 By”? dy -— - f0 ( 02:2 + 8.92 )dy (9.22) . I II y’ 2 , II 2 II 2), 3V(:1:,y,8) : 3V(:v.y ,8) _/ (0 Witty ,8) + 5‘ Way ,8) dy". 6y’ 6y” y,,:0 0 61:2 632 Integrating once more 31 V I / Wdy' = V(.r, y, s) — V(.r, 0, s) (9.23) 0 631' y H :> V(:c,y, s) = V(:I:,0,s) +/ w dy' — (9.24) 0 0y y":0 y ”I 52V($.y",8) 32V(Iay",8) ” , f0 /0 ( (9x2 + 032 )dy dy, (9.25) 195 we obtain a fixed point problem for V(;r,y,s). In the DA picture it converges to the exact solution in [71/2] steps, where n is the order of computation [98]. Since the Laplace equation is a second order PDE, we need two initial conditions. One is immediate from [y 6V(:1:,y”, s) 0 6y” dy' = yBy(;r, 0, s), (9.26) yu:0 because by definition BV/By = By. This is already known, and the other initial condition to be calculated is the potential in the midplane V(a:, 0, s). In the ideal case the potential in the midplane is computed by a path integral, along an arbitrary path. This is the case when the curl of the initial field is exactly vanishing. Due to various causes previously mentioned this is almost never true. Nevertheless, the curl it is usually small. Then, one should use a path along which the field is deemed more accurately computed, yielding a potential, and subsequently field components that are close to the original, and curl that is vanishing. Hence the name Maxwellification. Most of the time it is not obvious where the fields are computed more accurately. Then, one could try different paths and choose the one giving the smallest change in the field components. Convenient choices of paths are along the sides or diagonal of a rectangle in the midplane with opposite corners at (0, 0) and (:13, s). In the midplane we have ell/(7") = BU") - (17", where F: (3:, 0, 3). Integrating, we get vm = m) + for B’ (F) . df'. (9.27) We can neglect the immaterial constant V(0), and integration along the sides in one direction gives V(:I:,0,s) :/ Bx(x',0,0)dx'+/ B,(:z:,0,s')ds’. (9.28) o o 196 Integration along the sides in the other direction gives V(:r, 0, s) = / B,(0, 0, s’)ds’ +/ Bx(:c’,0, s)d;r'. (9.29) o o For integration along the diagonal we set 7" = AF with A 6 [0,1]. Then, (11" = Fd). and 1 1 V(:r,0,s) = :r/ Bx(/\£L‘,O,/\S)d)\+ s/ B,(/\:r,0,)\s)d/\. (9.30) o 0 One can check by direct calculation that indeed Bx(a:, 0, s) = W and B,(a:, 0, s) = flggo—“l. This completes the Maxwellification procedure. Once we have V(:c, y, s) we can compute the field components satisfying Maxwell’s equations in the whole region of interest by mere differentiation. In DA, the field components are computed as local Taylor expansions, so the method provides a local Maxwellification. That’s why, beside choosing the right path, it might be useful to average over a certain region to decide which approach is the best. Finally, it should be obvious how to extend all the equations in the case of full 3D Maxwellification, if originally the field components are given in all 3 variables (3:, y, s). For example, (9.28) is extended as a: s y V(:r, y, 8) :2/0 BI(.I", 0, 0)d:r' +/O B3(1:, 0, s')ds' +/0 By(:1:,y', s)dy', (9.31) and in the same way in other cases. In this situation, of course, there are many more path choices and no fixed point transformation of the Laplace equation is needed. As an example, in Table 9.5 we present the 3 component of the curl of the Maxwelli- fied field of Table 9.3. We mention that the now we present the results up to order 12 in the curl and the first non-vanishing element occurs at order 6. Also, notice the improvement in comparison with the curl in Table 9.4. 197 (V x B) after Maxwellification Coefficient;I Order Exp. (:13, s, y) -.4547473508864641E-12 6 2 0 4 -.1455191522836685E—10 7 2 1 4 0.3051757812500000E—04 8 6 0 2 -.2328306436538696E—09 8 4 2 2 -.3051757812500000E-04 8 2 0 6 0.1862645149230957E-08 9 4 3 2 -.1862645149230957E—08 10 6 2 2 -.1490116119384766E-07 10 4 4 2 -.1490116119384766E—07 10 2 4 4 -.1455191522836685E-10 10 2 0 8 -.2842170943040401E—13 11 5 0 6 0.1490116119384766E—07 11 4 1 6 ~.3552713678800501E—14 11 2 0 9 0.5960464477539063E—07 12 2 2 8 Table 9.5: The 3 component of the curl, up to order 12, after Maxwellification of the field in Table 9.3. 9.3.2 Global Maxwellification We saw that local Maxwellification is possible based on a closed magnet model consist- ing of line currents. One might imagine cases when the magnet model is not closed, and for some reason it is practically impossible to close it, or the actual closings change the original fields significantly. For such cases the S-Dependent Differential Algebraic Analytical Fourier Transform (SDDAAFT) of subsection 9.2.2 provides a way for global Maxwellification and minimal modification of the original fields in a neighborhood of the optical axis. The only drawback compared to the previous method is that we need the field computation of the unclosed model in all 3 variables (11:, y, s), which implies increased computer time. We start with the magnetic field vector B(:1:,y,s) representing the field of an unclosed magnet model computed using Biot-Savart law. Therefore, V - B = 0 and V x B ¢ 0. Then, there exists another vector B(:1:, y, s), which stands for the field 198 generated by fictitious line currents representing the closings of the model. Obviously, if is not unique, due to infinitely many possibilities of closing. It follows that V - B = 0 and V x (B+ B) = 0. Taking the cross product V x V x (B + B) = v (V- (E + R')) —A (B'+ R”) = 0, we obtain A (B,- + 17,-): 0; B, and R,- being the components in cylindrical coordinates of B and B. Now, we know from the general theory what is the structure of a function in cylindrical coordinates that satisfies the Laplace equation. Hence, we get 00 12,-(13¢, 3) = Z (f,,1(r, s) sin 1gb + 9,1(1', s) coslgb) 7" — B,-(r, (b, s). (9.32) 1:0 Apparently, we get the smallest R, in the vicinity of the optical axis if we choose the free parameters in f,,,(r,s) and gi,l(r, s), the true multipoles, such that they cancel the corresponding terms in B,-. This way we fix uniquely the true multipoles, that are anyway the dominating part, and let R,- to contribute only for the pseudo-multipole parts. Here we define as being a true multipole of order I the s—dependent function that is the coefficient of 7‘“1 cos [(1) or 7"” sin 13, respectively, in the expression of B,-. This definition makes sense, since it reduces to the usual definitions in the case when the fields are derivable from a magnetic scalar potential. Once the principle is understood, in practice we do not need to calculate explicitly R,. It is enough to have B, and extract the relevant terms, the true multipoles, then the out of axis expansion is performed, the potential is built up and the new fields are computed. The new fields will satisfy Maxwell’s equation; hence the name global Maxwellification. Still, one thing remains to be proved. The solution is really unique if we prove that the true multipoles are invariant with respect to which component of the original field We choose, B, or B¢,. This result is easily obtained in case we impose the vanishing curl and divergence conditions, as has been shown in the direct method. It can be 199 shown that this is also the case without imposing any constraints on the coefficients. This is the subject of section 9.4. All the methods described have been implemented in the DA based code COSY INFINITY [83, 84, 24]. 9.4 Structure of B, and B), for Non-Maxwellian Fields We start with the Cartesian components Bx($1y18) = Z aij(8)$iyja i,j=0 BALI/,8) = 2511(8)?ij 131:0 (9.33) (9.34) as given by the field computation, without imposing any relations among the aij’s and b,j’s due to Maxwell’s equations; 11. is the order of computation. Transformation to cylindrical coordinates gives B, = B, cos¢+ Bysin 35, B), 2 —B1 sing/1+ By cos 35, :1: = rcosgb, y = rsinefi. Inserting (9.33) in (9.35) we obtain 71 Br(r, (b, 3) = Z r’“ cosi ¢>sinj ab (a,j(s) cosrb + b,j(s) sin (15), i,j=0 n B¢(r, (b, 3) = Z r’” cos’ absinj ab (—a,-j(s) sin (b + bij(8) cos 95). 231:0 200 (9.35) (9.36) (9.37) (9.33) (9.39) (9.40) The next step is to transform the products of trigonometric functions into a sum of trigonometric functions involving multiple angles, i.e. i+j+1 +1 (bsinj n = Z a)” cos k3 + 1353’sin kg), (9.41) . . i+j+1 _ ' cos' 055m”1 (1) = Z 7):) cos kgb + (5),” sin kqfi, (9.42) 1:20 where ok’ ’,Bkj’ , 7): ’,6k” are real constants depending on j Hence, we obtain B i i+j 913(3) 2:?1 (a): (3’) cos k4) + 130" sin 19¢) + (9 43) r = 7' . . , . 14:0 519(5) 2:10“ (7,, (3’ cos kd) + 6”) sin k3) " . . —a,: (s) ”j“ 70’ cos k3 + 6“" sin k9) + 1' B¢ : 2 74+] 3 :6: (:1 2C.) 1:) ) W . (944) 1,):0 bij(3) Sic-:0 3 0k] cos kg) + (31] sin 10¢) Now we can use the definition of the multipoles to retain in B, and B4, just the true multipoles: we need to keep only the terms with k = i + j + 1, all others giving rise to pseudo—multipoles. We neglect the solenoidal terms, which are always treated best separately from 3,. Then, the components of the field containing just the true multipoles are a,j(s) (0523.“ cos (i +j + 1) 05 + Bfffilsin (I +j + 1) (15) n B, = E 1“” ”:0 +b,-,(s) (751,.“ cos (1' + j + 1) <9 + 6,3,.“ sin (2' +j + 1) as) B i H], -a,-j(s) (qgi’j+lcos(z+j+1)¢+6 S+1j+lsin(i+j+1)¢ ¢> : T . a +1.9) (e532,... cos 0' + ,- + 1) e + 9113,39) (1+ J + 1) e 201 or by rearranging terms the results are Br: 2 1,]:0 17. i,j=0 ("a«jj(3 ) n . . [- (a,j(s)a£i’j _ + (41.331313... + 1).-.3932...) sin (1 +1 + 1) e (j) +1 + b1j(3)’71+j+1)cos(i+j+1)¢ ( ) J) 8[7113+] + bij(8 )0 i+j+1) _ + (—a.,-(e)5§{3,+1 + 5,,(e)3§1’,,,) sin (I: +j + 1) e cos(i+j+1)q§ . (9.45) By expanding the trigonometric functions in terms of exponentials, it can be seen that: for j even we get C! (j) _ 1+j+1 " BEQj-H = 01 793'“ = 01 651’... = 2 “HM—1):”. and forjodd 0521'“ — 01 (3‘1in 2—(i+j) (_1)(j-1)/2’ 78))“ : _2—(1T+j)(_1)(j-1)/2 (SEQJ‘H : 0- 3 (9.47) (9.43) (9.49) (9.50) (9.51) (9.52) (9.53) (9.54) Separation of the double sum into summation over 2', and j-even, respectively j—odd leads to n 27.1 . ri+j “15(3) COS“ + j +1)¢)+ B. : Z 3:01—even bij(SSi ln(’l +1]. + 1)¢ r 2': 2’1 . ri+j 12(3) CO 5(2 +3 +1)¢)+ 1:1 J—odd aij S) sin(i + j +1)¢ .. D . b.-.-(s)(1+1 +119)- 19 _ Fora... a0(3) sin(i +j + 1)9 ¢ 1:0 Z" . ri+j “11(3) COS (i +j + IMH- J:1 J-odd 1),-(5 )sin(1 z+j +1) 0 2 10") 0 -0.00002» « _9 -2 10 1 —0.00004 . _4,10-9. -9_ —0.00005 ‘6'1°_9 -8-10 1 -0.00008 A - - * -1-10‘8> A - . . -300 -200 ‘100 0 100 200 -300 -200 —100 0 100 200 (b) Normal multipoles: b2(s), b6(s), b10(s), and b14(s). 100 0.02 80' ‘ 0.01» J//\\ 601 1 O v/\ 40’ . —0.01» 20’ 1 —0.02» . . . -0.03 - . - ~ . - 0 -300 —300 _100 0 100 200 -300 «200 ~100 0 100 200 [\ ' ‘9’ ’ 0 5 10 —0.00002 . 4,10-9. —0.00004 -9 2-10 -0.00005» j/“\k 0 -0.00008 , I .1: - r * ),/ —300 —200 -100 0 100 200 -300 -200 -100 0 100 200 (d) Skew multipoles: 02(3), (16(5), a10(s), and (114(3). Figure 9.3: Multipole strengths of the lead end as functions of s. 207' 10000 1. 0.2 8000) 0.1 5000 0 4000» 4 _0_1. 2000» —0.2» ‘ _ _ . —0.3» ‘ - - - _ ~200 —100 0 100 200 —200 -100 0 100 200 0 - - YA‘ - . . 0 n - —o.00002 _2,10—9 \[ , —0.00004 _4.10‘9» 1 -0.00006 _6.10-9. —0.00008» —8-10‘9 -200 -100 0 100 200 -200 -100 0 100 200 Figure 9.4: Normal multipole strengths of the return end as functions of s: b2(3), 06(8), 010(8), and 014(8). been closed at “infinity” (in practice meaning far away from the observation points). The result is contained in Figure 9.14. It can be seen that, although the agreement is pretty good for the quadrupole strength, the small differences are amplified for the higher order derivatives. 9.6 Computation of Fringe Field Maps As an application of the multipole decompositions, we describe very accurate high- order map computations of s—dependent fields. There are two ways to calculate maps. In the first case the following three steps are needed: using the analytical magnet model, the field expansions at selected support points along the optical axis are computed. In case it is necessary, the Maxwellification is included in this step. Then follows the extraction of the multipoles. Finally, the multipoles are interpolated by Gaussian interpolation [99], and using the integration algorithm of COSY, described 208 4 106» 2 101° « 10 1-10 2 1061 4 0 <_‘ /\ f\_ 0 T -1.101° V 6 —2-101°' -2 10 > ‘ -3-101° 6 L . - . . A . . . - . ‘4‘10 -300 -200 -100 0 100 200 -300 —200 -100 0 100 200 5 10": V ' ' V 7 V ‘ 1.5 1019 - - 19 4.10“» 1 1018 5-10 2-10“ I) 0 4AA v* A 1 .n V O ' V" -5-1018> VV -2-101" -1.1019» 19, -4 101‘) -1 5 10 —300 —200 -100 0 100 200 —300 -200 -100 0 100 200 24 r v 1 10 4.102, 5 1023 A 1 2.1023 h 1 28, -5.1023 —2-10 A . . -4-1028 - . - 1 —300 -200 —100 0 100 200 -300 -200 —100 0 100 200 Figure 9.5: The even order s—derivatives of the lead end’s b2(s). Shown are b2(2’(s), 52(4)“), 52“”(3), 52“”(3), 520043), and 5209(3). 2CH) 2000» 1500» 7 1000 1'10 ’ ‘fl 0 VA -j‘ Av 500» 1 V V' —1-107 0 J‘v N 7r -500» \/J M 1 ‘2 10 U -3 10’» 1 -1000 -300 -200 —100 0 100 200 —§00 —200 -foo 0 100 200 12 7 f V Y - 7 - . 7 1-10 . ‘ 1-1016 J 11 510 ’L ( O ‘1 11’ n- U U 1 o - - .1 l V 11 -1 1015» _5.10 > 1 16 —2 10 . _]_.1012 . 16 —300 —200 -Ioo 0 100 200 —300 —200 —100 0 100 200 Figure 9.6: The even order s-derivatives of the lead end’s b6(s). Shown are b6(2’(s), b6(4’(s), b5(6’(s), and b6(8’(s). 0.3 ' 10000» ' 0.2 5000 0.1 0 VA .A V AVA 0 'AvA T‘V M Auk -0.1 )1 -0.2 -5000- -0'3 ‘ 10000 —§00 —200 —100 0 100 200 ‘300 '200 ‘100 0 100 200 Figure 9.7: The even order s-derivatives of the lead end’s b10(3). Shown are b10(2’(s) and 010(4)(8). 21C) 100000 ~ r - v - 6-108 8? 50000» 4.103 -2-1o8 —SOOOO> -4.108 —6-108 ' ~1ooooo» ‘ -8-108 -300 -200 -100 o 100 200 —§00 -200 —100 o 100 200 1 1013 l 2 1017’ 17 12 p > ’ 0 AVAVAVM V' O VAVAVAVA A _ 1 . 1017 > V V“ 17 —2 1o . —5-10”» V 17 -3 10 ' —300 —200 —100 o 100 200 -§00 —200 —100 0 100 200 ’ 1-1027 ' , 1.1022 l 26 7.5-10 5-1021 5 1026’ 26 O ‘AvAvAv V [VA' 2 n 5 1 O n- -5.1021 l 1 22 v" ‘n V "A —2.5-1o 4.1022 45.1026 -1 5 1022 1 1 1 A -7.s-1026 1 . _ - -300 -200 -100 0 100 200 -300 —200 -100 o 100 200 Figure 9.8: The even order s—derivatives of the lead end’s a2(s). Shown are a2(2)(3), a2(4)(s), a2(6)(s), a2(8)(3), a2(1°)(s), and a2(12)(s). 211 1001 ' ' ' ' ' ‘ 6 v - v . 7 1.5-10 50, 1 1-106 J/\ 500000 1“ [\A A1 0 l/JL 0 AVVV l] vv \N/’ —500000 -50» ‘ ‘1'106 —1.5.106 - A —100' - . - ‘ —300 -200 -100 0 100 200 -§00 —200 —100 0 100 200 4.1010. 1 5.1015 15 2 1010 1'10 AL 5 101‘* / 0 AA“ A AAA - AAA VV' V V" 0 ‘UV'U ‘ 14 10 -5 10 —2 10 15 10 -1 10 ‘4 10 ’ —1.5 1015 —6-1010 _300 -000 -100 0 100 200 -300 —200 —100 0 100 200 Figure 9.9: The even order s-derivatives of the lead end’s (16(5). Shown are a5(2)(s), 0.5(4)(s), a6(6)(s), and a6(8)(3). 0.3 V 10000» 0.2 v 5000» 0.1 A A N —0.1 [ —0.2 -5000» ‘0'3 ‘ 10000 -3‘00 -2‘00 -1‘00 (3 150 200 " -300 -200 -100 0 100 200 Figure 9.10: The even order s-derivatives of the lead end’s (110(3). Shown are a10(2)(s) and alol4)(s). 3212 100» ' ' ' ' ' 1 . - 1 15106 50» 1-106 500000 All 0 AU VJN 0 AV[\‘_’L\IA V VAV~ -500000 '50' ‘ -1-106 -1.5-106 -100. - 1 A A A . 1 _ _ A _ A 1 . _300 _200 _100 0 100 200 300 200 100 o 100 200 4.101°’ « 1.5-1015 15 2.1010 1'10 AL 5 10“ m n O AAM A 0 AAA A AAA 1. U 10 V -S-10 u -2-10 15 10 —1-10 "4'10 ’ —1.5-1015 Figure 9.9: The even order s-derivatives of the lead end’s a5(s). Shown are a6(2)(s), 06(4)(s), 06(6)(s), and 06(8)(s). 0.3 T, f W 100001 ' ' ' ~ v 1 0.2 5000 1 0.1 -0.1 l1 -0.2 -5000» _0'3 i 10000 _3b0 -250 -100 5 150 200 _ —300 -200 -100 0 100 200 Figure 9.10: The even order s—derivatives of the lead end’s a10(s). Shown are 010(2)(s) and 010(4)(s). f212 4-101° 41061 21010 A l 6 2 10 > 0 /\ k 0 V 6 -2 101°’ -2-10 ' 6 —4-101°r -4-10 A A A A A - 1 A -200 -100 0 100 200 -200 —100 0 100 200 6-10“ 7 19 1-10 4-10“ < 13 5-10 2-10“* A l 0 AVA V 41 A .1 \l 0 v -5.1018 V _2,1014, « -1-1019 4140141 t-1.5-1019 H * -200 -100 0 100 200 ‘200 '100 0 100 200 14024, 7 . v ‘ 3-1028 1 . - H 28 2~10 5-1023 A A] 1-1028' 0 A L Vl V 0 J] All A lr'Ukl 23 -5-10 ’ -1.1023 J - - - —2-1028 . A -200 ~100 0 100 200 —200 -100 0 100 200 Figure 9.11: The even order s-derivatives of the return end’s b2(s). Shown are b2(2)(s), b2<4>(s), b2(6)(s), b2(8)(3), b2(1°)(s), and b2(12)(s). 213 2000 r ~ v ~ . 1 v V 1500’ 4 7D 1000» ~ 1’10 AA 0 J) 500- J/\ , ll bf? 7 0 -1'10 ’ -500» X/\j U/r 1-2-107’ —200 -100 0 100 200 -200 -100 0 100 200 12_ 11, O 1 NHL i 1.101: vMan 11 5.1011. V «-1-10161 ”U ll, -1.1012> i -200 —100 0 100 200 —200 -160 0 100 200 Figure 9.12: The even order s-derivatives of the return end’s b6(s). Shown are b6(2)(3), b6(4)(s), b6(6)(s), and b6(8)(s). 0.2 , 7500’ D 5000 0.1 2500 0 v “— 0 1 V V —O.1 ‘ —2500' —0.2. , -5000» -7500» J *0-3 A A A A A ‘ —10000 A 1 ~4 A -200 -100 0 100 200 '200 ‘100 0 100 200 Figure 9.13: The even order s-derivatives of the return end’s b10(3). Shown are b10(2)(s) and b10(4)(s). \1 200 100 ‘ °°°° - - 6’ 8000 4x10 . ‘5°°°°’ 2x105» 6°°° 45000:: ‘°°° "00°00 -200 400 \ 100 200 2000 425000 -2x10° 450000. a -200 -100 100 200 4"") Figure 9.14: The lead end’s normal quadrupole strength, and its first and second 3- derivatives, extracted from Maxwellian and non-Maxwellian fields, respectively, and superimposed on the same pictures. 214 Order Opening aberrations (:r|a") n Return end Lead end 2 O 0 3 0.2497609620388847 0.2995410193258450 4 0.6293863605053251E—13 -.3054844395050998E—13 5 0.1903231022089257 0.1962270889999381 6 0.6911276300083234E-12 -.6125473066348172E—12 7 0.1699451387437771 0.9019387229287767E-01 8 -.1364626459777126E—09 0.2013346179046971E-09 Table 9.6: Opening aberrations, (110"), for the return and lead ends’ exit focusing maps. in [24], the map is generated. The alternate way’s first step is the same. However, the scalar potential at sup- port points is anyway computed in the process of Maxwellification. We can use this potential to integrate the equations of the motion, with an interpolation scheme that preserves the derivatives at the support points, yielding the map. Both methods have been implemented in COSY, and they give essentially the same results. Especially at high orders, the Gaussian method is faster due to the smoothing properties of Gaussian interpolation [99]. Although to list the whole map it would be too long, to get a feeling of the resulting fringe field maps for the above mentioned end regions we list the opening aberrations in both ends up to order 8 in Table 9.6. Once we have the maps, they can be employed for dynamics studies, which is actually the final purpose of the whole theory and methods developed in this chapter. We applied the methods to study the fringe field effects in the LHC. Some of the results are presented in chapter 10. 215 Chapter 10 Fringe Field Effects on the Nonlinear Dynamics of the LHC Recently, many studies have been published on single particle dynamics in the Large Hadron Collider to be built at CERN. See [100] and references therein. Specifically, at collision energy it has been shown that the dynamics is dominated by the interaction region’s high gradient quadrupole triplets. This is due to large variations of the [3 functions across the quadrupoles. The studies concentrate on many possible realiza— tions of the LHC lattice, that is, on computing the effects of the so—called systematic and random body errors, and perhaps other effects like crossing angle, beam-beam interaction, misalignment, etc. The fringe fields, traditionally, are taken into account at most at the level of lumped thin lenses (kicks), characterized by integrated multi- pole strengths. However, it is not obvious whether this simplistic approach is enough to give an accurate account of the dynamics under the influence of fringe fields. This chapter attempts to fill this gap, by assessing the impact of fringe fields via very ac- curate fringe field maps, at least for the LHC, and specifically for the low-B insertion quadrupoles shown in Figures 9.1 and 9.2. We showed in chapter 9 how to take into account the local structure of s-dependent fields, resulting in the computation of very accurate fringe field maps. We use the 216 fringe field maps computed with the methods of section 9.6 to study systematically their effect on the nonlinear dynamics of the LHC. Starting from a simplistic model of the lattice, we gradually include more and more effects (described in more detail in the next section) and study their influence measured by tune-shifts, resonance strengths and sometimes resonance webs. We keep an eye on the importance of the exact shape of fringe fields and relative importance with respect to body errors. Results concerning off-energy particles are included too. 10.1 Methods of Analysis We use as measures for the dynamics indicators that have been proven effective in predicting the performance of existing accelerators, and were useful for construction of correction schemes for proposed machines. We will employ tune footprint, tune shift, resonance strength and resonance web calculations. All of them are based on normal forms of symplectic maps. We assume that the LHC is accurately described by the n-th order Taylor expansion of the system’s true map, M. Because hadron accelerators can be regarded as large Hamiltonian systems, the truncated map will be symplectic to order n. It means that relative to some symplectic coordinates 2' its Jacobian, M, satisfies the symplectic condition to order n MTJM = J. (10.1) The truncated map can be subjected to an order by order symplectic change of variables that finally yields its normal form. That is, there exist symplectic maps .An such that N = AnOM o A;‘, (10.2) The symplectic map N takes a particularly simple interpretation; it is a rotation with radius dependent frequency. See [101] for details. The angles of advancement. of a 217 point on a torus after one application of the map are called the tunes of the respective particle. Its deviation from the tune of a particle with 0 amplitude (the linear tune) is called the tune-shift. Due to the fact that the normal form of a symplectic map is unique, also the tune—shifts are uniquely defined. We compute these quantities for the various cases studied. In general, for good performance of accelerators, large tune—shifts are to be avoided. In this setting, the resonance condition is defined as I; - [1' (J13) = m (mod 27r), (10.3) for a vector of integers I: and integer m. The tunes are denoted by [1, f represents the action variables (the radii), and if the parameters. Although the normal form transformation can be used to compute the tunes only in the non-resonant case, it is our hope that extrapolation of the results close to the resonant case can give insight into the dynamics of resonant orbits. For this purpose, we study (10.3) as a function of .7. By adding to the right hand side a small quantity 8 << 1, with fixed maximum value, we plot in action space the actions j that satisfy the resonant condition. This gives insight into the resonance orbit structure of the phase space directly. This picture of the location and width of resonance lines is called the resonance web. The intuitive interpretation of the role of s is that it translates a fixed, small distance in tune space around exactly resonant orbits into oscillation of the action variables around exactly resonant orbits. The amount of oscillation gives a measure of the width of the resonance lines. Overlapping of resonance lines are considered signs of chaos, which is not necessarily bad in theory, but it is usually in practice. Often, the dynamic aperture is close to the chaotic boundary. Hence, the closeness of the resonance lines to the origin in action space again can be a useful indicator. It is not straightforward to explain the method of computing the resonance strengths 218 without getting into unnecessary details. This is partly due to the fact that the reso- nance strengths are not defined uniquely. Usually it is based on the normalizing map, An, which is known to not be unique. Moreover, the manipulation of the elements of .An is somewhat arbitrary. Nevertheless, as a general rule, the resonance strengths are directly proportional to the nonlinearity of the underlying map and its distance from resonances, measured by the resonance denominators, which are the left hand sides of (10.3). For more details of a method similar in spirit to our calculation we refer the reader to [102]. The difference between the two methods is that we do not use Lie algebraic techniques, but we work directly with the Taylor expansions of the components of the normalizing map. Despite all this, the correction of dominating res- onances proved to be an effective tool for improving the performance of accelerators. We could say that, although the exact numerical values of the resonance strengths do not really have relevance (as they are method dependent), the qualitative picture it is useful (for example identifying the dominating resonances). We mention that none of the above indicators have an absolute correlation with the behavior of particles in accelerators. In some cases one of them can have a better correlation with the dynamic aperture, in other cases another indicator, or none. However, altogether probably they can reveal the gross features of the dynamics, and are useful in practice. 10.2 Cases Studied All maps, fringe field and rest of the lattice (LHC v.5.1), have been computed at order 8 using the code COSY INFINITY [24]. Due to sensitivity to numerical er- rors, eSpecially at high orders, of the tune—shift computation, and occurrence of large numbers in the maps, we performed all the calculations in quadruple pre- 219 cision on an Alpha workstation. The effects of RF cavities have been neglected, i.e. we computed the maps in two transversal degrees of freedom, with energy as a parameter. At the specific location of the lattice where we fixed the Poincaré section, the r.m.s. beam sizes are 01, = 1.267 ~ 10‘4m and 0,, = 2.981 - 10‘4m, respectively. It is also known from the design specifications that the r.m.s. nor- malized emittance is EN = 3.75 . 10‘6m . r and the r.m.s. energy deviation is 01.3 = 1.1 - 10’“. For body errors we used the table MQXB (FNAL High Gradi- ent Quad) Reference Harmonics at Collision v. 2.0, available on the W at the address http://www.agsrhichomebnl.gov/LHC/fnal/v2.0/hgq..col_v2p0.txt. The de- tailed analysis of the body errors were not the main purpose of our studies, so we used only one seed for the random body errors, which gives “average” results in some sense (for example, dynamic aperture). For the systematic part of the errors we employed their full uncertainty, with the two possible signs. That is, if we denote by (0,.) the average value of a multipole, then the multipole value due to uncertainty lies between —d (2),.) + (0”) 3 bn S +d (bn) + (bu) . Hereafter we will refer to the two cases of full uncertainty by their sign, (-) or (+). To assess the importance of fringe field shape, we use two different models. One is the “detailed” shape, based on the exact shape of the fringe field computed using the model HGQSOI [92]. The other is a “default” fringe field shape, as implemented in the code COSY Infinity, based on the falloff modeled by an Enge function [24]. The detailed fringe fields detune the ideal lattice and also introduce linear coupling between the planes. To obtain meaningful results we have to retune and decouple the lattice. We achieve this in a rather elegant way using an ideal local correction. Moreover, the method provides a way to keep the design linear lattice completely 220 unchanged. It is done by splitting the fringe field maps into two parts Mn = Lff + NH» (10-4) where L f f is the linear part and N f f is the nonlinear part. Application of the inverse of the linear map, which can be thought of as the zero length insertion ideal local corrector, we obtain for the fringe field map MffZI-l-L-f-flofo, (10.5) where I is the identity map. The identity as linear part ensures that the linear layout of the lattice remains unchanged. Of course, we would get a slightly different result if we were to apply L}; from the right. However, L f f is close to identity, so L}; is also close to identity, and hence almost commutes with the nonlinear part. Again, this is an ideal case, and it is very likely that any real world correction scheme would introduce more nonlinearities in the map of the system. This method has been implemented in COSY INFINITY as a new fringe field mode. As a final remark, we mention that it is enough to compute only the exit focusing fringe field maps, and obtain the other variants by mirroring operations and rotations. Also, we use the same symmetry based tricks to get the correct maps for the proposed layout of the interaction regions, which includes rotations of quadrupoles around their vertical axis. The respective procedures are described in section 10.4. Table 10.1 contains all the cases studied, starting from the simple to the more complex. In the following section we describe the results obtained for each of them. We will refer to the specific case by their number in the table. 10.3 Results and Discussion The results shed light on the relative importance of intrinsic nonlinearities of the ideal lattice, the fringe field induced nonlinearities, and body errors induced nonlinearities. 221 [ CASE 1 SYSTEM 1 Interaction regions at order 8, rest linear Fringe fields and body errors OFF 2 Interaction regions at order 8, rest linear HGQ detailed fringe fields ON 3 Interaction regions at order 8, rest linear Only quadrupole components of HGQ fringe fields ON 4 Interaction regions at order 8, rest linear HGQ default fringe fields ON 5 Whole lattice at order 8 Fringe fields and body errors OFF Whole lattice at order 8 6 detailed fringe fields for HGQ, and default fringe fields for rest of ring ON; body errors OFF 7 Whole lattice at order 8 fringe fields OFF, body errors (—) ON 8 Whole lattice at order 8 fringe fields OFF, body errors (+) ON 9 Whole lattice at order 8 fringe fields from case 6 ON, and body errors (—) ON 10 Whole lattice at order 8 fringe fields from case 6 ON, and body errors (+) ON Table 10.1: The various LHC realization cases studied. 222 The tunes are visualized in two different ways. The 2D pictures represent the usual tune footprint style, and the 3D pictures show the tune shift of particles as a function of initial amplitudes in geometric space, in units of r.m.s. beam sizes, up to 60. The tune-shifts in the 3D pictures are all in units of 10‘4. The resonance strengths are computed close to the expected dynamic aperture, along the diagonal in action space. The units are arbitrary, and we denote I; = (q, p). Also, for cases 9 and 10 we computed the resonance strengths on a grid of points to identify the dominating resonances in different locations of phase space. Every case is subdivided in 3 subcases according to energy: 6 = -2.50’E, O, +2503. For the computation of resonance webs we used a maximum value of 5 = 10‘3. The size of the beam at the expected dynamic aperture is approximately J; = Jy = 5 - 10’4m. We start with the most simple case. Case 1 represents the linear LHC lattice with only the intrinsic nonlinearities of the interaction region quadrupoles added. As we expected, Figures 10.1, 10.2 and 10.3 show that the nonlinearities are insignificant. The tune footprints have regular shapes, but practically vanishing in size. For off-energy particles the dominating resonance is (2,0), which is 2 orders of magnitude bigger than the dominating resonance for on-energy particles, (2, -—2). Case 2 is actually case 1, to which we add the detailed fringe fields of the High Gradient Quadrupoles of the interaction regions. The acceptance guidelines require a tune shift of less than 10’3 at 60. We can see that the nonlinearities introduced by the fringe fields are considerable. See Figures 10.4, 10.5 and 10.6. However, tune shifts are still inside the safe region. Also, the tune footprints get bigger and have highly irregular shapes. Maximum values of tune shifts and size of tune footprints decrease as the energy increases. The sharpest decrease with energy is experienced by particles with small initial y and large 223 d=—2.5 s 0.334429 0.320059 0.334429 0.320058 0.334428 0.320058 0.334428 0.334427 0'320057 0.324671 0.324673 0.324675 0.210061 0.310062 0.3100(3 d=0,+,—2.5 s 0.30589 0.305889 0.305889 0.305888 0.296116 0.296116 0.3 0.305 0.31 0.315 0.32 0.325 Figure 10.1: Tune footprints for the LHC case 1. Figure 10.2: Tune shifts for the LHC case 1. 224 3.0-10’9 I" t u, I .’ I 2.010-9 ‘ 1 I u; 2' I I6? 05“. i,’ a; 5" ’I 2 I I, I I I '0 Q I. 7 18: v 0‘ II ’1 UL) 1.0-10" '5 7"? lf‘i' ‘01": 1‘0? 52%: 0"?! I 4;? In“! ’t I 1' I; I "02‘: D '0 so ’: Figure 10.3: Resonance strengths for the LHC case 1. initial :0. Dominating resonances are the (2, -2), also encountered in case 1, and a newly excited resonance, the (1, —-1). There is an increase in the absolute values of the resonance strengths, compared with case 1. The resonances are almost invariant with respect to energy. Also, their magnitudes are slightly increasing with energy, an opposite tendency when compared to tune shifts. Next, we studied what is the relative importance of the quadrupole components with respect to the rest of the multipoles. Therefore, case 3 is case 2 without the sextupole components in the fringe fields. For the results, see Figures 10.7, 10.8 and 10.9. We notice a 5 times decrease for the maximum tune shift for negative energy dis- persion particles, and some more modest decreases for the other particles. The tune footprints are becoming a little bit smaller and regular, triangle shaped, hence the irregularities of case 2 are caused by the sextupole components in the fringe fields. Overall, the importance of the sextupole components are decreasing as the energy increases. There are noticeable differences between resonance strengths of cases 3 and 2. The magnitudes drop by 2 orders of magnitude, and become comparable with off-energy particle’s resonance strengths of case 1. Moreover, they are not invariant anymore with respect to energy. While the dominating resonance, regardless of en- 225 0.33625 / 0.3362 0.33615 // 0.3361 ////// 0.33605 ////// 0.3265 0.3266 0.3267 0.3268 d=2.5 s . .-°"/ 0.30424 --",./ 0.30423 -------- 'fi// 0.30422 '.--“AAA./// 0.30421 /.// // 0.3042 ¢ 6 0.30419 ¢ 0.30418 Jll' ' 0.30417 0.29442 0.29443 0.29444 0.29445 0.29446 d=—2.5 s 0.3201 0.3201 0.320 0.3200 0.3200 0.3200 0 31004 4 2 1 8 6 4 0.31008 0.31012 0.31014 d=0,+,— 2.5 s 0.305 , 0 .295 0.3 0.305 0.31 0.315 0.32 0.325 Figure 10.4: Tune footprints for the LHC case 2. 6 Figure 105: Tune shifts for the LHC case 2. 226 Figure 10.6: Resonance strengths for the LHC case 2. ergy, is (2, —2), which we carried along since case 1, again (2,0) from case 1 makes its appearance for off-energy particles, and (1, ——1) completely disappears. We con- clude that the intrinsic nonlinearities of the quadrupoles excite the (2,0) and (2, —2) resonances, all components of the fringe fields contribute to the (2, —2) resonance, making it the dominating one, and the (1, -1) resonance is excited only by the sex- tupole components, which is comparable in magnitude with (2, —-2). The magnitudes of the resonance strengths hold the slightly increasing behavior with energy. Now we turn our attention to the question of the exact shape of the fringe fields. For easy comparison we created case 4, which is case 3 with the detailed fringe fields, containing only the quadrupole components, replaced by a generic fringe field shape, that has a falloff modeled by an Enge function with 6 parameters. As one can see, the differences between the set of Figures of case 3, Figures 10.7, 10.8 and 10.9, and the Figures of case 4, Figures 10.10, 10.11 and 10.12, are marginal. The pictures are almost identical, with a small decrease in all the indicators for the generic fringe field. We conclude that, at least for the main component of the fringe fields, the exact shape is not critical. As the next step, we studied the nonlinearities of the ideal lattice, that is no errors at all, only the intrinsic nonlinearities of the whole ring up to order 8. This is case 5, 227 d=—2.5 s 0.3201 0.33628 0.32009 0.33626 0.32008 0.33624 0.32007 0.33622 0.32006 0.32652 0 32653 0.32654 0.32655 0.32656 0.3 006 0.31007 0.31008 0.31009 0.3101 d=2.5 s d=0.+,- 2.5 5 0.335 0.30419 0.33 0.304185 0.325 0.32 0.30418 0.315 0.304175 0.31 0.30417 0.305, 0.29445 0.29446 0.29447 0.29448 0_295 0.3 0305 0_31 0.315 0_32 0.325 Figure 10.7: Tune footprints for the LHC case 3. Figure 10.8: Tune shifts for the LHC case 3. 228 Figure 10.9: Resonance strengths for the LHC case 3. and the relevant Figures are 10.13, 10.14, 10.15 and 10.16. The effects of the intrinsic nonlinearities of the whole lattice are comparable with those created by the fringe fields of the interaction regions. In the case of 6 < O the fringe fields cause a bigger maximum tune-shift, while for 6 2 0 the intrinsic nonlinearities are marginally bigger. Also, the tune footprints are roughly the same, but in this case the shapes are regular, triangle shaped. However, the resonance strengths are more than one order of magnitude smaller than in case 2, and they are almost invariant with respect to energy. Many more resonances are excited than in the previous cases, with (2, —-2) remaining the dominant one. Other excited resonances are (1,2), (1,0) and a few smaller: (1,—2), (3,0) and (2,0). The (2,0) is almost completely missing for 6 = 0, hence it is excited mostly by off-energy particles. The resonance web shows the chaotic boundary to be at approximately 3 - 10‘3m. A beam of 9: 1202.17 occupies 2 4 — 5 . 10‘4m in this picture, therefore this region is completely free of low order resonances. We estimated the following resonances to be the “thickest”, in decreasing order: (1,—1), (2,—2), (3, —3), (4, —4), (6,1) and (8, —1). Case 6 represents case 2 and case 5 superimposed, and additionally generic fringe fields are set on for the remaining of the ring. 229 0.310065 0.310075 0.310085 0.31009 d=0,+,- 2.5 s 0.32009 0.33449 0.320085 0.33440 ‘ . 69 0.32008 0.33447 @KKL. 0 320075 . ..a(-‘ - 0.33446 KKKK-gdufil Qg§§714““‘ 0.32007 0‘33‘45 0.320065 0.33444 0.32006 0.32468 0.32469 0.3247 0.32471 d=2 . 5 s 0.33; 0.30591 0‘33 0.305905 0'325 0.32 0.3059 0.315 0.305895 0.31 0.30589 0.296115 0.296125 0.296135 0.29615 0.3 0.305 0.31 0.315 0.32 0.325 Figure 10.10: Tune footprints for the LHC case 4. Figure 10.11: Tune shifts for t 230 he LHC case 4. Figure 10.12: Resonance strengths for the LHC case 4. Figures 10.17, 10.18 and 10.19 show that the tune shifts and footprints of case 6 are almost like adding up the corresponding pictures from cases 2 and 5. The resonance strengths are looking similar to those of case 2. This proves that the dominating fringe field effects are concentrated in the interaction regions, which has been showed to hold for other effects too, which are limiting the dynamic aperture. The next two cases deal with the body errors. As mentioned in section 10.2, we have two possible signs for the uncertainty part of the errors. We use the same “average” seed for the random part in both cases. Case 7 is the (—) case, and case 8 is the (+) case. For case 7, Figures 10.20, 10.21 and 10.22 show a little bigger tune shifts and footprints than for case 6, but still inside the safe region. The footprints are elongated and curved, and even overlapping for 6 = —2.5aE and 6 = 0, respectively. On the other hand, the resonance strengths are amplified by 3 orders of magnitude. The new dominating resonance is (0,3). The resonances that are excited mainly by fringe fields and intrinsic nonlinearities are negligible. Case 8 is the first situation where the maximum tune shift exceeds the acceptable level of 10‘3 at 60 (Figure 10.24). See also Figures 10.23 and 10.25. Comparison of cases 7 and 8 proves that the systematic errors are important. 231 0.3196 0.319575 0.31955 0.319525 0.3195 0.319475 0.31945 0.32062 0.32061 0.3206 0.32059 0.32058 0.32057 d=-2.5 s ..IHHHHH 0.30992 0.30996 d=2.5 s m ll ll .11nIImmmmmmumu 0.31096 0.311 0.32008 .,” d=0 0.3200. ”mm” 0.32004 ”mu” 1. '4 0.32002 1’ 1%,; ’1 0.32 0.31990 _ .30998 0.31 0.31002 0.31006 0.3215 d=0,+,- 2.5 S ‘ ‘ 0.3095 0.3105 0.3115 0.3125 Figure 10.13: Tune footprints for the LHC case 5. Figure 10.14: Tune shifts for the LHC case 5. 232 ~ ~a~ .. .... ~ ~ ~ ~ ”.75 - --3 - # a“ s ~ : i: ~ _ . .......$-- 3 tests so ‘9 § Hr. so. .0 I‘ m. . e: : ~ .6 a} 4 ~ 6 :fimuwvs ava 0. 11 .3 #.6 . 6 e «5...... ...... a Figure 10.15: Resonance strengths for the LHC case 5. .. ...4 .. 4,... ..7 u 1 .., ... if i! 0.0005 0.001. 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0025 0.002 .0015: 0 0.001: 0.0005- =0. Figure 10.16: Resonance web for the LHC case 5, for d 233 1.9'10'5 vévfi 2. ... 9 a .4 -~ -~ s $.wie: 5 ...qsm.» .éfir =u.-ai‘"§ l» VI . 4. u. 3.. afifiae ii fibfi‘ .56."..."505 1. 13%? a 3““. 555 ~ gflv :.~H. . “is". § ~ -n-w~7a Figure 10.15: Resonance strengths for the LHC case 5. .0035 0.004 “13.3' v; ‘3‘ 0 157‘ if ‘ |\'\I F 0.0005 0.001 0.0015 0.002 0.0025 0.003 0 =0. Figure 10.16: Resonance web for the LHC case 5, for d 233 Figure 10.19: Resonance strengths for the LHC case 6. However, notice that only a small fraction of the total particles, with predominantly horizontal initial amplitude, have tune shifts bigger than 10‘3. The other tune shifts are still bigger than in case 7, but within acceptable limits. The tune footprints are regular, triangle shaped, very elongated in one direction. The dominating resonance is still (0,3), with an important contribution from (1, 2). Despite the fact that the tune shifts are bigger than in case 7, the magnitude of the resonance strengths are slightly smaller in case 8. The resonance structure is invariant with respect to energy. The last two cases are the closest to reality among all studied cases. We included the whole lattice at order 8, detailed fringe fields of the High Gradient Quadrupoles in the interaction regions and generic fringe fields for the rest of the ring, and body errors for interaction regions with the two possible signs. So, case 9 is actually cases 6 and 7 superimposed, and case 10 is cases 6 and 8 superimposed, respectively. Figures 10.26 and 10.27 show that the tune footprints are only slightly distorted compared to case 7. The only case when the fringe fields clearly have a considerable effect is for 6 < 0 particles, but still inside acceptable limits. From Figure 10.28 it is obvious that the dominating resonances are excited by the body errors. Hence, th e body errors dominate the fringe field effects. Note that the fringe fields alter the natural chromaticity (even change its sign). It is interesting to compare Figures 235 0.3201 0.32005" 0.3198 0.3196 0.3194 0.3215 0.3206 0'321 0.32058 . 0-3205 " 0.32 0.32056 0.3195 0.32054 0.319 0.3108 0.3i09 0,311 o_3111 0.3095 0.3105 0.3115 0.3125 Figure 10.20: Tune footprints for the LHC case 7. Figure 10.21: Tune shifts for the LHC case 7. 236 Figure 10.22: Resonance strengths for the LHC case 7. 10.29 and 10.16. The resonance channels in this case move much closer to the origin, inside the region occupied by the beam at target dynamic aperture. The chaotic boundary is around the target dynamic aperture. The thickest resonance channels remain (1, —1), (2, —2), (3, —3), (4, —4), (0,3) and (1,2). Beside the fact that the tune shifts for 6 < 0 are much bigger in case 10, Figures 10.30, 10.31 and 10.32 tell the same story as case 9. Again, the situation resembles case 8, only in this case the fringe fields have a little more influence than they had in case 9. We computed the resonance webs for 6 = —2.503 and 6 = 0 to show a general conclusion, that the resonance web structure is essentially invariant with respect to energy. This is consistent with the invariance of resonance strengths with energy. From Figure 10.33 results that the closest resonance lines are slightly farther away from the origin than in case 9. Also, in general the magnitudes of the resonance strengths increase with energy, and the resonance strengths of case 10 are smaller than the resonance strengths of case 9. It is exactly the opposite for the tune shifts; the biggest are of case 10. Finally, for cases 9 and 10 we computed the resonance strengths for 6 = 0 on a grid in action space, to identify the dominating resonances at different locations. Due 237 d=—2.5 s 0.3196 0.3194 0.3192 0.319 0.3188 0.32005 0.32 0.31995 0.3199 0.31985 0.3198 '. 0.31975 O.3201~ \ ‘ § 0.30975 0.31025 0.31075 d=2.5 s 0.3115 3.3098 0.31 0.3102 0.3104 d=0,+,— 2.5 S 0.31C 0.3205 0.32025 0.32 0.31975 0.3195 0.31925 0.319 0.31875 \ \. R 0.3108 0.3109 0.311 03112 03 0975 0.31025 0.31075 Figure 10.23: Tune footprints for the LHC case 8. Figure 10.24: Tune shifts for the LHC case 8. 238 0.3115 6 :9 I 1? i" ,4 '71 I? I’ Figure 10.25: Resonance strengths for the LHC case 8. to lack of space, only the picture that summarizes the conclusions is shown in Figure 10.34. Obviously, (0,3) is the dominating resonance in a major part of action space. For exactly horizontal motion (3, O) is the dominating one, and there is a narrow region of predominantly horizontal motion for which (1, ——1) for case 9, and (1,—1) or (1,2) for case 10 are the dominating resonances, respectively. As we saw, the (1, —1) is excited by the quadrupole component of the fringe fields. Hence, there is a small region where the dominating resonance is given by fringe fields. However, the magnitudes of the respective resonances are very small. Actually, the magnitude of the resonance strengths increase much more quickly in the vertical direction. Also, from the resonance web pictures results that the closest resonance line to the origin is always along the vertical. In a final paragraph we draw a few general conclusions. While the fringe fields generate important dynamical effects, as far tune shifts, footprints and resonance strengths are concerned, they are dominated by body errors. If there is a correlation between dynamic aperture and these quantities, than also the DA is determined by body errors. The fringe fields do decrease the dynamic aperture of the ideal lattice by a factor of around 6, as was shown in section 4.5. In this sense, they have a 239 d=—2.5 5 6:0 0.32015 _ 0.322 0.3201 0.3218 0.32005 0.3216 0.32 0.3214 - 0.31995 0.3212 0.3117 0.3119 0.3121 0.3123 0-3098 0.31 0.3101 0.3103 d=2.5 s d=0,+,— 2.5 s 0.31892 -_ 0.3189 0.31888 0.31886 V 0.31884 0.3101-17 - —. 0.30915 0.30925 0.30935 0-30945 0.3095 0.31 0.3105 0.3110.3115 0.312 Figure 10.26: Tune footprints for the LHC case 9. Figure 10.27: Tune shifts for the LHC case 9. 240 ’0 0,. :0 10,; O I, 9,0,0 0 I '1 Iii/"II t“U" I h ¢ ‘0 9 I I" O O .0 .0 ‘9 0,. o u. h '0 O l l I '20:? 1'”! .Qqé of)” Figure 1028: Resonance strengths for the LHC case 9. “\— 1&9» .1 (:Av’“v if. 1 - e/ $202252; 0.0004 #’ / \~'::‘:~-'-‘§A.. Va 0.0003 QR" 0.0002 / v I, f :/ . ‘é;;lj;7%§£g if. '; r" ," , “41"}, ;-‘\ 0.0001 - //(/7"{7¢11f{r 3’; r”::,'{,~‘;;‘:&' “m 0’ M 2:? l .- 1. . me- A- - 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 Figure 10.29: Resonance web for the LHC case 9, for d = 0. 241 =—2.5 s 0.32u4 . 0.3214 0.3201 0.3212 0.32 0.321 ‘3'"; '1=> fa 0.3208 é 0.3199 ‘dd ’ / 0.3206 ’4 0.3198 ’I . . 0.3204 ' ' 0.312 0.3125 o_313 0.3135 o_314 0.3098 0.31 0.3102 0.3104 0.3106 0.3108 d=2.5 s d=0,+,— 2.5 3 0.321: ‘3. 0.31895 0.321 0.3205 -.,. 0.3189 0.32 \ 0.31885 ._ 0.3195 0.3188 1’._ 0-319\ 0.2091 0.3092 0.3093 0.3094 0.3095 0.3096 0. 09 0.31 0_311 0.312 0.313 0.314 Figure 10.30: Tune footprints for the LHC case 10. Figure 10.31: Tune shifts for the LHC case 10. 242 Figure 10.32: Resonance strengths for the LHC case 10. huge effect. However, this reduced DA is still way above the expected DA; the approximate DA value with fringe fields is around 200. However, the body errors alone decrease the DA to the level where is starts to hit a lower acceptable limit, and the fringe fields do not affect it very much anymore. Indeed, this is confirmed by tracking; the loss in DA due to fringe fields is at most 05%0- On the other hand, the DA cannot be correlated exactly with both tune shifts and resonance strengths. This can be seen from the fact that we do not get the largest tune footprints for the case with the largest resonance strengths and vice versa. However, roughly it is correlated with both indicators. The resonance strength and resonance web results are consistent in the sense that resonance lines closer to the origin give larger resonance strengths. Regarding the shape of fringe fields, we could conclude that the exact shape does not matter. However, intrinsically the fringe field effects are not very important for the LHC, and for other situations, like the proposed Muon Collider, where the relative importance of the fringe fields are greater, the shape of fringe field is also important. For the LHC, most of fringe field effects are generated by the High Gradient Quadrupoles. The energy dependence of the tunes show that in general tune shifts are maximized for predominantly horizontal motion for 6 < 0, and as energy increases, the maximum decreases and shifts towards predominantly vertical motion. For on—energy particles the tune shifts are approximately symmetric with 243 O O O O O O 0005/ .. ‘~’ . "" / \‘E?1\\\\in 7”,? .0004 .0003- .0002- .0001- 0.0001 0.0002 0.0003 . ,_ 5%?f' . .9 '4 ,,, -- = .. 5:42;?3-9‘ \t -.‘ , , 7.1- "*1, ;Eé;»;’.4r:/A?« \«\ «‘0' .../”Kw /l' . \ . 5112.087 :flé/i‘\\\ A\\\ {33.7 . L x) ' . V V ‘ \A‘Q‘Up ... '.. . IF 4' 0.0001 0.0002 0.0003 Figure 10.33: Resonance web for the LHC case 10, for d = —2.531.; and d = 0. 244 (0,3) 2-104 -‘- a w I a I i " ’(1,-1)or(1,2) 0 I (3,0) 5104 ..L Figure 10.34: Dominating resonances for the LHC cases 9 and 10. 245 respect to diagonal, with minimums attained around the diagonal. This implies that, if the tune shifts are correlated with the DA, the estimation of the DA using only on-energy particles launched along the diagonal will result in an overestimation of the DA. Resonance strengths and resonance webs are roughly invariant with respect to energy. However, a marginal increase of the resonance strengths with energy is usually observed. The trend for the tune shifts is exactly opposite. In some cases, a substantial decrease of the footprints is observed as energy increases. Over a large portion of action space the dominant resonance is (0,3). The resonance strengths increase in magnitude faster in the vertical direction. The magnitudes of the dominating resonances in the strip of predominantly horizontal motion are much smaller than in the rest of action space. The biggest jump in the magnitudes is observed at around 2 - 10‘4 along the vertical, which, by coincidence or not, is the location of the closest resonance line to the origin in the resonance web pictures. 10.4 Map Transformations Under Orientation Flips Sometimes, orientation changes of particle optical elements in an accelerator lattice are used as a mean to compensate or correct nonlinearities, improving the character- istics of the accelerator. One specific case is the LHC HGQs in the interaction regions [103]. The question arises what are the proper symmetry operations on certain maps that give the correct answer for maps of systems that differ by some symmetry trans- formation. That is, knowing the map of an element in a “default” orientation, what kind of transformations are necessary in order to obtain the maps of the “flipped” elements? Specifically, we need 3 different (similarity) transformations on the map, corresponding to mirroring the “default”, or “forward” element with respect to a plane. Here we use W. Wan’s nomenclature [104]. Mirroring the forward element with respect to the a: — y plane gives the “reversed” element, with respect to the y — z 246 plane gives the “switched” element, and with respect to the a: - z plane gives the “upside-down” element. In [104] it is explained how to obtain the map of mirrored elements knowing the map of the forward element. The map of the reversed element in the (3:, a, y, b, l, 6) symplectic basis is given by MR: ROM—10R, (10.6) where {100000\ 0—10000 001000 R” 0 0 0 —1 0 0 (10'7) 000010 \00000—1) is an involution: R o 72 = I, I being the identity. It follows that R“: ’R. The map M is the map of the forward element. Analogously, the switched map can be obtain from the forward map as M5=SOM08, (10.8) with f —1 0 0 0 0 O \ 0 -—1 O 0 0 O O 0 l 0 0 0 5‘ 0 0 0100 (109) 0 O 0 0 1 0 k 0 O 0 0 0 1 ) possessing similar properties as 72: S o S = I, and S“: S. The last one is the upside-down transformation MU=U0M0L1, (10.10) 247 with {100000\ 010000 00—1000 u_ 00 0 _100 (10.11) 000010 \000001} Again, U o 11 = I, and LI "1: LI. It is well known that an element satisfying midplane symmetry is invariant under the upside-down transformation. Next, we show under what conditions an element is invariant with respect to reversion. The conditions can be deduced from Hamilton’s equations. In the case of planar reference orbit, no electric fields, and s-independent magnetic fields (in which case the fields are derivable from the vector potential component A,, and AI = Ag 2 0), the most general Hamiltonian [105] is invariant under the transformation R. Moreover, the canonical equations of motion, and hence the map of a such an element, are invariant under the transformation 720., where a means that it inverts the sign of the independent variable 3, when acting on a map: ROOM = M 0 R0. (10.12) In the Lie Algebraic notation, the map can be written in the form M = 6“”; (10.13) where : H : is the Poisson bracket operator attached to the Hamiltonian of the system. Hence, 01(M ) = M "1, since : H : commutes with itself. Inserting it in the invariance relation we obtain that ”ROM“ = M072 (1014) .M = ROM—10R. (10.15) Comparison with the map of the reversed element gives the result In the specific case of the LHC HGQs, 180 degree rotations of quadrupoles around the y axis are performed. This transformation is equivalent to mirroring the forward element with respect to the :1: — y plane, and then with respect to the y — 2 plane. In other words, we obtain a “combined” element as the combination of the reversed and switched element. Finally, we can identify the combined element’s map with the map of the “opposite” quadrupole’s map. For details, see [103]. Now we can show that the combined element is unique, that is the reversed and switched, and switched and reversed elements have the same map. i.e. the two symmetry operations commute. 1) Reversed and Switched MC: 80 (ROM—10R) 08. (10.17) 2) Switched and Reversed M’C -_—. Ro(soM05)‘lo7z (10.18) = RosoM-losok (10.19) = soRoM-lokos (10.20) = 50 (ROM—1072) 03 (10.21) = MC. (10.22) We used the fact that S ‘1 = S, and being diagonal matrices [12,5] = ROS —S 0R 2 0. Actually, all the individual transformations commute due to the fact that they are generated by diagonal matrices. 249 So we have an easy procedure to compute the maps of arbitrary “opposite” el- ements from the “default” ones. Moreover, the map approach is valid for arbitrary field configurations, including detailed fringe fields and compositions of several maps. In the case of s-independent elements, we obtain simplifications due to invariance under reversion. This property is relevant for the implementation of the body errors in section 10.3, in which case we can get the map of the Opposite quadrupole by switching the default map. Of course, in the case of detailed fringe field maps, one needs to perform the reversion too. To compare the above results with the rules for multipole sign changes due to orientation changes [103], we derive the results based on field multipole expansion. Suppose that (10.23) gives the multipole expansion in the default reference frame. B, + 1B,, = 2(1),, + 1a,) (.2: + iy)". (10.23) As a contrast with the map methods, this formula, and hence the derivation, is valid only for s-independent elements, or the integrated strengths in the end regions. In the same way, we assume that the expansion in the opposite frame is . I , r I , r 11 By! + szt = Z ((3,, + tan) (:1: + 2y) . (10.24) n First, the reversion consists of the following transformations: :1." —> a: y' —> y z’ —> —2, (10.25) BI; —> —B,, By: ——> —B,,:. (10.26) Inserting them in the expansion gives B, +13, = Z — (0;, + ia'n) (1: +131)" . (10.27) Comparing coefficients with (10.23) we obtain that b E -b,,, (10.28) a E —a,,. (10.29) 250 The switching can be cast in a similar way to correspond to the following trans- formations :r' —> —:c y’—>y z'—+z, B... —> B, B,.—>—B,,, _.B,, + 1B, = Z (0;, +1a’n) (—x + 1y)". 11 Taking the complex conjugate of the above expression we arrive to —By — 18,, = Z (b;I — 20;) (—3: — 13))" 11 3,, + 18,, = Z(—1)" (—b;, + 10;) (:1: + 1y)". 11 Comparing coefficients with (10.23) we obtain 11’ a (—1)"+lb,,, a E (—1)"a,, The upside-down transformation can be expressed as I .7: —+ :1: y’-—>—y z'——>z, BI! —) —Bx ByI—-)Byl, B, - 1B,, = Z (0,, +15”) (1: — 1y)". 11 Taking the complex conjugate expression we arrive to By .1.in = :01," —z'a;,) (.1: + 1y)", n which gives the relation between multipole coefficients, 0, E 0.", a E —a,,. 251 (10.30) (10.31) (10.32) (10.33) (10.34) (10.35) (10.36) (10.37) (10.38) (10.39) (10.40) (10.41) (10.42) Finally, we get the rule for the combined (opposite) element as the commutable product of the reversion and switching transformation 0" III A | p—l V :3 Q“ 3 (10.43) a s (—1)"+1a,,. (10.44) in agreement with [103]. Direct calculation shows that the above results are in com- plete agreement with [103] for every case, after, in case it is necessary, the polarity is changed following the transformation such that the fundamental term remains posi- tive. As a conclusion, the above map manipulations based on the matrices R, S, and U form a complete set of commutable transformations to perform any flip scenario, including arbitrary s-dependent fields, and maps of composed elements. All the nec- essary map computations, compositions and inversions are easily performed in the Differential Algebra based code cosy INFINITY. 252 Chapter 11 Summary of Part II Accurate simulation of fringe field effects requires more effort than the traditional piecewise arclength independent fields. While some approximations and simplifica- tions work in some cases, there is the danger that one always relies on them, and as a consequence, yield inaccurate or misleading results in situations were fringe fields cannot be safely neglected anymore. It was shown that, in general, fringe fields af- fect the motion at all orders, starting with the linear part of the transfer map, and induce modifications in every indicator used to asses the single particle dynamics: center tunes, Chromaticities, tune shifts, resonance strengths, and dynamic apertures. Moreover, it is not straightforward to asses a priori the importance of fringe fields, and it is recommended to deal with fringe fields on a case-by-case basis. It was shown that fringe fields are important “in absolute value” for the LHC, but not a limiting factor, are a limiting factor for the dynamic aperture of the Neutrino Factory, and in- duce only a very slow diffusion for the Proton Driver, which does not alter the short term dynamic aperture. Further work identified that only a few matching section quadrupole fringe fields are responsible for the drastic reduction of the DA of the Neutrino Factory, and increasing their length, with the simultaneous reduction of the strength, restored the DA above the target value [12]. More detailed fringe field and other nonlinear effects simulations of the Muon Accelerators can be found in [56]. 253 In the end, careful consideration of fringe field effects amounts to integration of the correct ordinary differential equations of the motion, in which the nonlinear effects are accounted for, including the electromagnetic fields in accordance to Maxwell’s equa- tions. Besides fringe fields, the so—called kinematic effect can also lead to inaccurate results [56]. It was shown that the sharp cutoff approximation leads to divergences in the map, truncation of high order pseudo—multipoles can also give inaccurate results, and not even symplectification can undo the errors made in modeling the systems, as, for example, neglect of details of the fringe field shapes. It was shown that it is possible to obtain very accurate fringe field maps for current dominated superconducting magnets, using Differential Algebraic methods. The procedure includes field computation of current wires in the Differential Algebra framework, employing a novel form of the Biot-Savart formula for rectilinear currents. The method was illustrated by the multipole decomposition and map computation of the LHC’s HGQ ends. Clearly, the method can be used unaltered for any element with large enough radius of curvature, and can be adapted for bending elements by using different coordinate systems. Moreover, it is also possible to obtain multipole information from measured field data, for example from data on the surface of a cylinder, or in one or several planes. All is needed is some good interpolation method, that interpolates both the field values and the derivatives accurately. Although more research is needed in this direction, the Gaussian interpolation is promising [106]. The need to satisfy Maxwell’s equations in the ODEs resulted in the development of two methods to enforce them. It was shown that, in the Differential Algebraic framework, it is trivial to check and enforce Maxwell’s equations locally as well as globally. These methods are useful to correct small computational errors or occasional magnet design flaws. 254 Finally, a detailed study of fringe field effects on the LHC single particle dynam- ics revealed that the fringe fields introduce large nonlinear effects, and decrease the dynamic aperture typically by a factor of six, but the resulting DA is still well above the target. The fringe field effects are concentrated in the interaction regions. It was shown that the fringe field effects are dominated by the HGQ body errors. How- ever, we mention that for the body errors the worst case scenario was utilized for the systematic parts of the errors, and the fringe field errors were neglected. In conclusion, fringe fields can be important for any accelerator, and there are methods that allow a careful and accurate representation and simulation of their effects on the dynamics. Once the fringe field maps have been computed, the effort pays off by noticing that the subsequent dynamical studies require the same effort with fringe fields as without, including normal form based quantities such as tunes and tune shifts, Chromaticities, resonance strengths and webs, and symplectic tracking for fast, efficient, and reliable dynamic aperture estimation. 255 Bibliography [1] J.M. Sanz-Serna and MP. Calvo. Numerical Hamiltonian problems. Chapman and Hall, London, 1994. [2] Z. Shang. KAM theorem of symplectic algorithms for Hamiltonian systems. Numer. Math, 83:477—496, 1999. [3] Z. Ge and J. Marsden. Lie-Poisson Hamilton—Jacobbi theory and Lie-Poisson integrators. Phys. Lett. A, 1332134—139, 1988. [4] G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near-to- the-identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys., 74, 5-621117—1143, 1994. [5] B. Leimkuhler, S. Reich, and R. D. Skeel. Integration methods for molecular dynamics. In IMA Summer Program: Molecular Biology, K. Schulten and J. Mesirov, Eds, New York, 1994. Springer-Verlag. [6] J. Wisdom, M. Holman, and J. Touma. Symplectic correctors. Fields Institute Communications, 10:217—244, 1996. [7] G. Benettin and F. Fasso. From Hamiltonian perturbation theory to symplectic integrators and back. Appl. Numer. Math, 29, 1:73-87, 1999. [8] M. Suzuki. Quantum Monte Carlo methods and general decomposition of ex- ponential operators and symplectic integrators. Physica A, 205:65—79, 1994. [9] M. Berz. Differential algebraic description of beam dynamics to very high orders. Particle Accelerators, 24:109, 1989. [10] M. Berz. Modern Map Methods in Particle Beam Physics. Academic Press, San Diego, 1999, also available at http: / / bt.nscl.msu.edu/paperscgi/display.p1 ?name=AIEP108book. [11] M. Berz, K. Makino, and B. Erdelyi. Fringe field effects in Muon Rings. In Proceedings of HEMC’QQ, Workshop on Muon Colliders at Highest Energies, Long Island, 1999. Montauk. [12] F. Zimmermann, C. Johnstone, M. Berz, B. Erdelyi, K. Makino, and W. Wan. Fringe fields and dynamic aperture in the FNAL Muon Strorage Ring. Technical Report SL-2000—011 AP and NUFACT-NOTE 21, CERN, 2000. 256 [13] M. Berz, B. Erdelyi, and K. Makino. Fringe field effects in small rings of large acceptance. Phys. Rev. S T—Accel. Beams,, 3:124001, 2000. [14] D. Abell. Analytic properties and approximation of transfer maps for Hamilto- nian systems. PhD thesis, University of Maryland, 1995. [15] J. Shi and P. Suwannakoon. Single-particle dynamics in particle storage rings with integrable polynomial factorization maps. Phys. Rev. E, 58, 6:7868—7873, 1998. [16] I. Gjaja, A.J. Dragt, and D.T. Abell. A comparison of methods for long-term tracking using symplectic maps. IOP Conf. Ser., 131:173—184, 1993. [17] M. Berz. Symplectic tracking in circular accelerators with high order maps. In Nonlinear Problems in Future Particle Accelerators, page 288, Singapore, 1991. World Scientific. [18] H. Hofer. On the topological properties of symplectic maps. Proc. Royal Soc. Edinburgh, Sect. A, 115:25-38, 1990. [19] R. deVogelaere. Methods of integration which preserve the contact transforma- tion property of the Hamiltonian equations. Technical Report 4, Department of Mathematics, Univ. Notre Dame, 1956. [20] P.J. Channell. Symplectic integration algorithms. Technical Report Internal Report AT-6, ATN-83-9, Los Alamos National Laboratory, 1983. [21] P.J. Channel] and J .C. Scovel. Symplectic integration of Hamiltonian systems. Nonlinearity, 3:231—259, 1990. [22] A.J. Dragt and D.T. Abell. Symplectic maps and computation of orbits in particle accelerators. Fields Institute Communications, 10:59—85, 1996. [23] A. J. Dragt, L. M. Healy, F. Neri, and R. Ryne. MARYLIE 3.0 - a program for nonlinear analysis of accelerators and beamlines. IEEE Transactions on Nuclear Science, NS-3,5:2311, 1985. [24] K. Makino and M. Berz. COSY INFINITY version 8. Nuclear Instruments and Methods, A427z338—343, 1999. [25] Y.T. Yan, P.J. Channel], and M. Syphers. Performance of an algorithm for symplectic implicit one-turn map tracking. Technical report, SSC Laboratory Report SSCL-157, 1993. [26] R. Warnock and J. Ellison. From symplectic integrator to Poincare map: spline expansion of a map generator in Cartesian coordinates. Appl. Numer. Math, 29:89—98, 1999. [27] H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, MA, 1980. [28] K. Feng, H.M. Wu, M.Z. Quin, and D.L. Wang. Construction of canonical dif- ference schemes for Hamiltonian formalism via generating functions. J. Comp. Math, 7, 1:71—96, 1989. 257 [29] K. F eng. The calculus of generating functions and the formal energy for Hamil- tonian algorithms. J. Comp. Math, 16, 6:481—498, 1998. [30] OK. Siege]. Symplectic geometry. Amer. J. Math, 65:1—86, 1943. [31] A. Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Ad- vances in Math, 6:329—346, 1971. [32] A. Weinstein. Lagrangian submanifolds and Hamiltonian systems. Ann. of Math, 98, 2:377—410, 1973. [33] A. Weinstein. Normal modes for non-linear Hamiltonian systems. Invent. Math, 20:47—57, 1973. [34] A. Weinstein. Lectures on symplectic manifolds. CBMS Reg. Conf. Series in Math. 29, AMS, Providence, RI, 1977. [35] R. Abraham and J .E. Marsden. Foundations of Mechanics. Addison-Wesley, Reading, Second edition, 1978. [36] J .P. Amiet and P. Huguenin. Generating functions of canonical maps. Helv. Phys. Acta, 53:377—397, 1980. [37] AM. Ozorio de Almeida. On the symplectically invariant variational principle and generating functions. Proc. R. Soc. Lond. A, 431:403—417, 1990. [38] M.J. Sewel] and I. Roulstone. Anatomy of the canonical transformation. Phil. Trans. R. Soc. Lond. A, 345:577—598, 1993. [39] Henri Poincare. New Methods of Celestial Mechanics, volume 3, chap. XXVIII. American Institute of Physics, New York, 1893/1993. [40] A. Weinstein. The invariance of Poincare’s generating function for canonical transformations. Invent. Math, 16:203—213, 1972. [41] A. Banyaga. The structure of classical difieomorphism groups. Dordrecht, Boston, Kluwer Academic, 1997. [42] A. J. Dragt. Lectures on nonlinear orbit dynamics. In 1981 Fermilab Summer School, New York, 1982. AIP Conference Proceedings Vol. 87. [43] R. Abraham, J. E. Marsden, and Tudor Ratiu. Manifolds, Tensor Analysis, and Applications. Springer-Verlag, New York, 1988. [44] B. Erdelyi, J. Hoefkens, and M. Berz. Rigorous lower bounds for the domains of definition of extended generating functions. in preparation, 2001. [45] D. McDuff and D. Salamon. Introduction to symplectic topology. Oxford Uni- versity Press, New York, 1995. [46] H. Hofer and E. Zehnder. Symplectic invariants and Hamiltonian dynamics. Birkhauser, Base], 1994. 258 [47] J. E. Marsden and T. S. Ratiu. Introduction to Mechanics and Symmetry. Springer-Verlag, New York, 1994. [48] P. Libermann and CM. Marle. Symplectic geometry and analytical mechanics. Dordrecht, Boston, Kluwer Academic, 1987. [49] I. Vaisman. Symplectic geometry and secondary characteristic classes. Prog. in Math., Birkhauser, Boston, 1987. [50] M. Gromov. Pseudo-holomorphic curves in symplectic manifolds. Invent. Math, 82:307—347, 1985. [51] G. Rangarajan and M. Sachidanand. Symplectic integration using solvable maps. J. Phys. A, 33:131-142, 2000. [52] C. Johnstone. Collider ring lattices. In Proceedings of HEM C ’99, Workshop on Muon Colliders at Highest Energies, Long Island, 1999. Montauk. [53] C. Johnstone. Large acceptance, transitionless lattice for FNAL Proton Driver. In Proceedings of the NuFact’00 Workshop, Monterey, 2000. MC Collaboration. [54] CM. Ankenbrandt et a]. Status of muon collider research and development and future plans. Phys. Rev. S T—AB, 2:081001, 1999. [55] RB. Plamer, A. Sessler, and A. Tollestrup. Feasability study of muon colliders. ‘ In Proceedings of Snowmass ’96, DPF/DPB Summer Study on High-Energy Physics, Menlo Park, CA, 1997. SLAC. [56] M. Berz, K. Makino, and B. Erdelyi. Nonlinear effects in muon accelerators. Phys. Rev. ST -Accel. Beams, submitted, 2000. [57] B. Erdelyi and M. Berz. Theory of extended generating functions for symplec- tification of truncated Taylor maps. Phys. Rev. S T-Accel. Beams, submitted, 2000, preliminary version available at http://130.83.124.174/frames/index.htm. [58] B. Erdelyi and M. Berz. Optimal symplectic approximation of Hamiltonian flows. Phys. Rev. Lett., in print, 2001. [59] G. Rangarajan, A. Dragt, and F. Neri. Invariant metrics for Hamiltonian sys- tems. In Proceedings 1991 Particle Accelerator Conference, San Francisco, CA, 1991. IEEE. [60] C. Altafini, The De Casteljau algorithm on SE(3). A. Isidori, F. Lamnabhi— Lagarrigue, and W. Respondek (eds. ), Nonlinear Control in Year 2000. Springer-Verlag, Berlin, 2000. [61] Y. Eliashberg and L. Polterovich. Bi-invariant metrics on the group of Hamil- tonian diffeomorphisms. Int. J. Math, 4:727—738, 1993. [62] A.J. Dragt. Lectures on nonlinear dynamics and Lie methods with applications to Accelerator Physics. Univ. of Maryland preprint, 1998. [63] M. Bialy and L. Polterovich. Geodesics of Hofer’s metric on the group of Hamil- tonian diffeomorphisms. Duke Math. J., 76:273—292, 1994. 259 [64] F. Lalonde and D. McDuff. Hofer’s L°° geometry: energy and stability of Hamiltonian flows. Parts I, II. Invent. Math, 122:1—33, 35—69, 1995. [65] K.F. Siburg. New minimal geodesics on the group of symplectic diffeomor- phisms. Calc. Var. and PDEs, 3:299—309, 1995. [66] C. Viterbo. Symplectic topology as the geometry of generating functions. Math. Ann., 292:685—710, 1992. [67] Dusa McDuff and Jennifer Slimowitz. Hofer-Zehnder capacity and length min- imizing Hamiltonian paths. arXiv:math.SG / 0101085. [68] Y. Long. Geodesics in the compactly supported Hamiltonian diffeomorphism group. Math. Z., 220:279—294, 1995. [69] J .C. Sikorav. Systemes Hamiltoniens et topologie symplectique. ETS, Editrice Pisa, 1990. [70] Z.J. Shang. 0n KAM theorem for symplectic algorithms for Hamiltonian sys- tems. PhD thesis, CAS Computing Center, Beijing, 1991. [71] G. Lee-Whiting. Nuclear Instruments and Methods, 83:232, 1970. [72] H. Matsuda and H. Wollnik. The influence of an inhomogeneous magnetic fring- ing field on the trajectories of charged particles in a third order approximation. Nuclear Instruments and Methods, 77:40, 1970. [73] H. Matsuda and H. Wollnik. Third order transfer matrices of the fringing field of an inhomogeneous magnet. Nuclear Instruments and Methods, 77:283, 1970. [74] H. Matsuda and H. Wollnik. Third order transfer matrices for the fringing field of magnetic and electrostatic quadrupole lenses. Nucl. Instr. Meth., 103:117— 124,1972. [75] B. Hartmann, H. Irnich, and H. Wollnik. Analytical determination of 5th-order transfer matrices of magnetic quadrupole fringing fields. In M. Berz, S. Martin, and K. Ziegler, editors, Nonlinear Problems in Accelerator Physics, pages 87—96, Bristol, 1993. Institute of Physics Publishing. [76] M. Venturini. Scaling of third—order quadrupole aberrations with fringe field extension. In Proceedings of the 1999 Particle Accelerator Conference, New York, 1999. IEEE. [77] E. Forest and J. Milutinovic. Leading order hard edge fringe fields effects exact in (1 +6) and consistent with Maxwell’s equations for rectilinear magnets. Nucl. Instr. Meth. A, A269:474—482, 1988. [78] E. Forest, D. Robin, A. Zholents, M. Donald, R. Helm, J. Irwin, and H. Moshammer. Sources of amplitude-dependent tune shifts in the PEP-II design and their compensation with octupoles. In Proceedings of the Fourth European Particle Accelerator Conference, London, 1994. World Scientific. [79] H. Wollnik. Optics of Charged Particles. Academic Press, Orlando, Florida, 1987. 260 [80] J. D. Jackson. Classical Electrodynamics. Wiley, New York, 1975. [81] W. Wan, C. Johnstone, J. Holt, M. Berz, K. Makino, M. Lindemann, and B. Erdelyi. The influence of fringe fields on particle dynamics in the Large Hadron Collider. Nucl. Instr. Meth. A, 427:74—78, 1999. [82] M. Berz. Arbitrary order description of arbitrary particle Optical systems. Nu- clear Instruments and Methods, A298z426, 1990. [83] M. Berz. COSY INFINITY, an arbitrary order general purpose optics code. Computer Codes and the Linear Accelerator Community, Los Alamos LA-11857- C2137, 1990. [84] M. Berz. Computational aspects of design and simulation: COSY INFINITY. Nuclear Instruments and Methods, A298z473, 1990. [85] M. Berz. COSY INFINITY Version 8 reference manual. Techni- cal Report MSUCL-1088, National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, 1997. see also http: / / www.beamtheory. nscl.msu.edu / cosy. [86] K. L. Brown and J. E. Spencer. Non-linear optics for the final focus of the single- pass-collider. IEEE Transactions on Nuclear Science, NS-28,3:2568, 1981. [87] D. Bazin. personal communication, 2000. [88] F. Meot. personal communication, 2000. [89] Ingrid Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992. [90] L. Sagalovsky. Beam transport optics of dipole fringe-field in the framework of 3rd order matrix theory. Nucl. Instr. Meth. A, 298, 1-3:205—222, 1990. [91] F. Zimmermann. Tune shift with amplitude induced by quadrupole fringe fields. Technical Report SL—2000-009 AP and NUFACT-NOTE 18, 2000. [92] G. Sabbi. Magnetic field analysis of HGQ coil ends. Technical Report TD-97- 040, Fermilab, 1997. [93] F. Méot. The ray-tracing code Zgoubi. Nucl. Instr. Meth., A 427 :353—356, 1999. [94] The LHC Study Group, The Large Hadron Collider, Conceptual Design. The Large Hadron Collider, Conceptual Design. Technical Report AC / 95-05, CERN, 1995. [95] F. Pilat et a1. US-LHC IR magnet error analysis and compensation. In Pro- ceedings of the EPA C, Stockholm, Sweden, 1998. IOP. [96] S. Russenschuck. A computer program for the design of superconducting accel- erator magnets. Technical Report CERN AT/95-39, CERN, 1995. [97] S. Caspi et a]. An approach to 3D magnetic field calculation using numerical and differential algebra methods. Technical Report LBL-32624, L. Berkeley Laboratory, 1992. 261 [98] M. Berz. Differential Algebraic Techniques, Entry in ’Handbook of Accelerator Physics and Engineering’, M. Tigner and A. Chao (Eds. ) World Scientific, New York, 1999. [99] K. Makino and M. Berz. Arbitrary order aberrations for elements characterized by measured fields. In Optical Science, Engineering and Instrumentation ’97, SPIE, 1997. IEEE. [100] J .-P. Koutchouk. The LHC dynamic aperture. In Proceedings PAC ’99, New York, 1999. AIP. [101] M. Berz. Differential algebraic formulation of normal form theory. In M. Berz, S. Martin and K. Ziegler (Eds. ), Proc. Nonlinear Effects in Accelerators, page 77, London, 1992. IOP Publishing. [102] R. Bartolini and F. Schmidt. Normal form via tracking of beam data. Technical Report LHC Project Report 132, CERN, 1998. [103] L. Bottura, A. Jain, D. McChesney, F. Pilat, G. Sabbi, and J. Wei. Summary of US-LHC magnet database workshop. Technical Report RHIC/ AP/ 160, BNL, Long Island, New York, 1998. [104] W. Wan. Theory and Applications of Arbitrary-Order Achromats. PhD thesis, Michigan State University, East Lansing, Michigan, USA, 1995. also MSUCL- 976. [105] K. Makino. Rigorous Analysis of Nonlinear Motion in Particle Accelerators. PhD thesis, Michigan State University, East Lansing, Michigan, USA, 1998. also MSUCL-1093. [106] K. Makino and M. Berz. Arbitrary order aberrations for elements characterized by measured fields. In Optical Science, Engineering and Instrumentation ’97. SPIE, 1997. 262