APPLICATIONSOFTHEDABASEDNORMALFORMALGORITHMONPARAMETER-DEPENDENTPERTURBATIONSByAdrianWeisskopfATHESISSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofPhysicsŒMasterofScience2016ABSTRACTAPPLICATIONSOFTHEDABASEDNORMALFORMALGORITHMONPARAMETER-DEPENDENTPERTURBATIONSByAdrianWeisskopfManyadvancedmodelsinphysicsuseasimplersystemasthefoundationuponwhichproblem-perturbationtermsareadded.Therearemanymathematicalmethodsinperturbationthe-orywhichattempttosolveoratleastapproximatethesolutionfortheadvancedmodelbasedonthesolutionoftheunperturbedsystem.Theanalyticalapproacheshavetheadvantagethattheirap-proximationisanalgebraicexpressionrelatingallinvolvedquantitiesinthecalculatedsolutionuptoacertainorder.However,thecomplexityofthecalculationoftenincreasesdrasticallywiththenumberofiterations,variables,andparametersconsidered.Ontheotherhand,thecomputer-basednumericalapproachesarefastonceimplemented,buttheirresultsareonlynumericalapproxima-tionswithoutasymbolicform.Anumericalintegrator,forexample,takestheinitialvaluesandintegratestheordinarydifferentialequationuptotherequestedstateandyieldstheresultasnumbers.Therefore,noalgebraicexpression,muchlessaparameterdependencewithinthesolutionisgiven.Themethodpresentedinthisworkisbasedonthedifferentialalgebra(DA)framework,whichwasdevelopedtoitscurrentextentbyMartinBerzet.al[3,4,5].TheusedDANormalFormAlgorithmisanadvancementbyMartinBerzfromthearbitraryor-deralgorithmbyForest,Berz,andIrwin[13],whichwasbasedonanDA-Lieapproach.BothstructuresarealreadyimplementedinCOSYINFINITY[18]documentedin[7,16,17].There-sultofthepresentedmethodisanumericallycalculatedalgebraicexpressionofthesolutionuptoanarbitrarytruncationorder.Thismethodcombinestheeffectivenessandautomaticcalcula-tionofacomputer-basednumericalapproximationandthealgebraicrelationbetweentheinvolvedquantities.ACKNOWLEDGEMENTSFirstofall,IwouldliketothankmyacademicadvisorProfessorMartinBerzforhiscontinuoussupport,patienceandhelpfulguidanceinmyresearch.Furthermore,IparticularlyappreciatedthediscussionswithDavidTarazona,RobertHipple,ErikaKazantseva,andRaviJagasiaduringmyworkandwouldliketothankallofthem.IalsoshallnotforgetScottPratt,WadeFisherandKyokoMakinoforbeingsokindandagree-ingtobeonmythesiscommittee.Thelastyearhasbeenagreatexperiencewhichwasonlypossibleduetothegenerousscholar-shipbytheStudienstiftungdesdeutschenVolkesandfurthersupportbyMichiganStateUniversityformystudiesandresearchhere.Iamverythankfulforthisopportunityandallthegreatmemoriesthatcamealongwithitduringtheyear.Lastbutnotleast,IwouldliketorecognizeKimCrosslan,whowasalwaysthereifquestionsorproblemsofanykindarose.iiiTABLEOFCONTENTSLISTOFTABLES.......................................viLISTOFFIGURES.......................................viiiCHAPTER1INTRODUCTION...............................11.1Motivation.......................................11.2Basicconcepts.....................................41.2.1FloatingPointnumbers............................41.2.2TruncatedPolynomialExpansion.......................61.2.3DifferentialAlgebra..............................71.2.4Diagonalization................................101.2.5Action-Anglecoordinates...........................141.2.6TheFlowOperator..............................161.2.6.1ExampleofsymmetricallyperturbedHamiltonian........17CHAPTER2INTEGRATORS................................202.1FlowIntegrator....................................202.2Fourth-OrderRunge-Kuttaintegrator.........................232.3FixedpointIntegrator.................................242.4Errorvsstep-sizeforRK4integrator.........................262.5Integratorcomparison.................................29CHAPTER3THENORMALFORMALGORITHM....................323.1TheDANormalFormAlgorithm...........................323.1.1Diagonalization................................343.1.1.1Diagonalizationtransformationofthelinearpart.........343.1.1.2Diagonalizationtransformationofthenon-linearparts......363.1.2Thenon-lineartransformation........................393.1.3TransformationtoNormalFormcoordinates.................44CHAPTER4PROTRACTINGCALCULATIONSINPERTURBATIONTHEORY....484.1Variationofparameters................................484.2Exampleofananalyticalperturbationtheoryapproach...............494.2.1Symmetricperturbedharmonicoscillatorexample..............494.2.2Asymmetricperturbation...........................55CHAPTER5PERTURBEDHARMONICOSCILLATOR.................575.1ThePendulum.....................................575.1.1IntroductiontotheProblem..........................575.1.2Unperturbedcase...............................595.1.3Pendulumtransformation...........................625.2NormalFormUniqueness...............................69iv5.3ComparisontoLieTransformperturbationTheory..................715.4SolvingperturbedHarmonicoscillatorwithperharmosc.fox.............79CHAPTER6CONCLUSION................................83APPENDIX...........................................85BIBLIOGRAPHY........................................88vLISTOFTABLESTable2.1Thetablerepresentstheq-componentofthetransfermapoftheinitialstatefromt=0totinthePendulumvectorfromequation5.26uptothe11thorder.ThetransfermapwascalculatedwithonestepviathethreedifferentDAbaseintegrators.ThecolumnswithboldordernumbersindicatetermsofO(t5),whicharetheerroraffectedtermsoftheRK4integrator,whichsupportsthetheorydiscussedinsection2.4.Furthermore,theresultsofedpointintegrationandFlowintegrationareidentical,whichalsoagreeswiththetheory.......................................30Table2.2Thetableshowsthep-componentequivalenttoTable2.1.............31Table5.1CoefoftheCOSYresultuptoorder10inrforthePendulumtuneswt0(r)showninequation5.28withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.........................65Table5.2CoefoftheCOSYresultforthePendulumPeriodTt0(q;p)showninequation5.33withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.................................67Table5.3CoefoftheCOSYresultforthetuneswt0(q1;p1)showninequa-tion5.59withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.Atablewithcoeftoequation5.59uptoorder10in(q1;p1)arelistedintheappendixintable5.4....................74Table5.4COSYcoefofwt0(q1;p1)uptoorder14forthecalculationinequation5.59..........................................75Table5.5Thetablerevealsthetermsandrelatedcoefofwt0(q0;p)fortheoscil-lationaroundtheedpoint(q00;p0)=q32+p32;0oftheHamiltonianinequation5.39withtheparameters(w0;e;a)=(1;2;2=9)andq0=qq0.....79Table5.6ThetableillustratesanexampleoutputofCOSYfortheperiodTinthe(q1;p1)-coordinatesupto5thorderin(q1;p1).Thecolumn'I'denotestherow-counter.Thecolumnsunder'EXPONENTS'eachrepresentonevariable.Thetwoarethe(q1;p1)-coordinates.Eachadditionalcolumndenotesaparameterdk;elorgistartingwiththeparameterassociatedwiththeper-turbationtermenteredtotheprogram.Thenumberintherespectivecolumndenotestheexponentofthevariable/parameter.Thecolumn'ORDER'sumsupalltheexponentsandpresentsthetotalorderoftheterm.Thesecondcolumn'COEFFICIENT'yieldstheCOSYTaylorexpansioncoefre-gardingtheassociatedterm.............................82viTableA.1COSYcoefofr2(q1;p1)uptoorder10in(q1;p1)forthePendulumcalculation......................................86TableA.2COSYcoefofwt0(q1;p1)uptoorder10in(q1;p1)forthePendulumcalculation......................................87TableA.3COSYcoefofwt0(q;p)uptoorder10in(q;p)forthePendulumcal-culation........................................87viiLISTOFFIGURESFigure1.1TheplotshowstheclassicRK4-trackingofthePendulum-ODE(eq.5.4)fromthe(-1,1)-initialstateuntilt=p=4.Allparametershadtobegivenvaluesandweresetto1..........................2Figure1.2Thepurplearearepresentsasectionoftheinitialstatesetinphasespace(q0;p0)=([1:3;1:3];[1:3;1:3]).ThegreenareadenotesthetransfermapM(q0;p0)oftheinitialstatesetinthePendulumvector(eq.5.4)attimet=0:5;1;2andp,respectively.TheDAbasedRK4methodwithastep-sizeofh=0:01wasusedfortheintegrationtoalgebraicallyconnectthewholeinitialstatesetuptoordermwiththestate.Thestatesetisnormallyextendedovermoredimensionswhichalsoinvolvestheparameter-space.Forthe2D-illustrationtheparametersaresetto1...................3Figure1.3IllustrationofphasespacecoordinatetransformationtoAction-Anglecoor-dinates........................................15Figure2.1ThegraphillustrateshowthetotalerrorofaDAbasedRK4-integrationdrops˘h4untilacertaincriticalstepsizeisreached.Fromthisstep-sizeonwards,morestepsdonotresultinmorepreciseresultssincethemaximumpreci-sion,determinedbythepointcalculationerrors,isreached.Forthisexample,thecriticalstep-sizeisapproximately210h0ˇ104h0.Theer-rorisestimatedbycalculatingthedifferenceoftheresultfromthestep-size2kh0totheresultfromthestep-size212h0,whichisconsideredthe'ex-act'solution.Thedifferenceisthennormalizedtotherespectiveresultofthe20h0-step-size-calculation.Thedifferentplots/colorsrepresentthecoef-ofdifferenttermsofthet=2h0mapofthePendulumODE(eq.5.4)aftertheintegrationfromthet=h0mapusingadifferentnumberofstepsaccordingtothestep-size..............................27Figure2.2Thegraphillustrateshowthesingle-steperrorofaDAbasedRK4-integrationstepdrops˘h5withhbeingthestep-size.Theerrorisestimatedbycalcu-latingthedifferenceoftheresultfromthestep-size2kh0totheresultfromthestep-size29h0,whichisconsideredtheexactsolutioninthiscase.Thedifferenceisthennormalizedtotherespectiveresultofthe21h0-step-size-calculation.Thedifferentplotsrepresentthecoefofdifferenttermsofthet=h0+hmapofthePendulumODE(eq.5.4)aftertheintegrationfromthet=h0mapusingonestepofstep-sizeh.Theadditionalfactorof25intheerrorto1/step-sizeratiooriginatesfromnormalizationwithrespecttothe21h0-step-size-calculation.........................28Figure5.1IllustrationshowsmathematicalPendulumoflengthlwithpointmassminagravitationalofstrengthg[10].......................58viiiFigure5.2TheshowsthephasespacecurvesofthePendulumoscillationaccord-ingtothetransfermapatt0=1,whichwasgeneratedbyintegratingequation5.4withtheDAbasedRK4in1000stepsofstep-sizeh=0:001.Fortheil-lustration,allparametersweresetto1.Thetransfermapwastrackedfor1000iterations.Thedifferentcurvesrepresentthefollowinginitialcondi-tions,listedfrominnertooutercurve:q=:3;:6;:9;1:2;:::;3:0;p=0.De-tailsregardingtheseeminglyclosedandfragmentedcurvescanbefoundinsection5.2......................................63Figure5.3TheshowsthephasespacecurvesofthePendulumoscillationaccord-ingtothetransfermapatt=1,whichwasgeneratedbyintegratingequation5.4withtheDAbasedRK4in1000stepsofstep-sizeh=0:001withordertruncation20.Fortheillustration,allparametersaresetto1.Thetransfermapatt=1wastransformedtoNormalformcoordinatesandtrackedfor1000iterations.Thedifferentcurvesrepresentthefollowinginitialcondi-tions,listedfrominnertooutercurve:q=:3;:6;:9;1:2;:::;3:0;p=0.Thecurvesshowcircularmotionandtheseparationinthefragmentedcurveshasaconstantdistanceincontrastto5.2.Detailsregardingthisareinsection5.2......................................64Figure5.4Thegraphillustratestheamplitudeq0dependenceoftherelativeperiod-errorfordifferentamplitudeshiftsDq..........................68Figure5.5Thephasespacecurvesoriginatefromthesamemapusedin5.2.Thedifferentcurvesconsistof'low'periodedpoints,whichrepresentthespe-resonancesofthecurve.Thedistancebetweenthesingleresonancepointsillustrateshowthefrequencychangesalongthecurve...........70Figure5.6Thephasespacecurvesoriginatefromthesamemapusedin5.3.Thedifferentcurvesconsistof'low'periodedpoints,whichrepresentthespe-resonancesofthecurve.IntheNormalForm,thedistancebetweenthesingleresonancepointsisconstantalongonecurve,whichmeansthatthefrequencydoesnotchangealongthecurve.TheNormalFormcoordinatesareuniqueinthisproperty.............................70Figure5.7ThegraphshowsthepotentialV=q22q42+q69oftheHamiltonianinequa-tion5.39fortheparameters(w0;e;a)=(1;2;2=9).Therearethreestablestationarypointsattheoriginandq=q32+p32.Thetwounstablestation-arypointsareatq=q32p32.Thepotentialallowsoscillationineachofthethreevalleysaswellasaglobaloscillationoverlargep............76ixFigure5.8ThetransfermapoftheODEinequation5.57wascalculatedusingtheRK4with100stepsofstep-sizeh=0:001untilt=0:1.Theillustrationshowsthephasespacetrackingof1000iterationsusingtheparameters(w0;e;a)=(1;2;2=9).ThecurvesaroundtheoriginaresimilartotheoneinthePendu-lumexamplein5.2.Forlargerjqjandsmallpthephasespacecurveoscillatesaroundadifferentedpointatapproximately(1:5;0).Forlargeqandpthephasespacecurveoscillatesaroundallthreeedpoints.Figure5.7makesthisbehaviorapparent..........................77Figure5.9Trackingof1000iterationsusingtheparameters(w0;e;a)=(1;2;2=9)ofthetransfermapusedin5.8inNormalForm.Incomparisonto5.8onlythecurvesaroundtheoriginarepreserved,whichillustrateshowperturbationtheoryonlyworksinthedirectsurroundingoftheconsiderededpoint......................................78Figure5.10Trackingpictureofthesametransfermapusedin5.8onlywithashiftedreferencepointfortheperturbation.....................78xCHAPTER1INTRODUCTION1.1MotivationManydetailedmodelsinphysicsarebasedonasimplermodel-anidealcase-wheretheexactsolutiontotheproblemisknown.Themathematicalmethodsofperturbationtheorycansubse-quentlybeusedtoapproximatethesolutiontotheadvancedproblemfromtheexactsolutionoftheidealcase.ExamplesofrealisticsystemsarethePendulumoscillationortheenergystatesofahydrogenatominanelectricBothcanbeapproximatedorevensolvedwithperturbationtheoryduetotheestablishedsolutionsoftheidealcases,namelythesimpleharmonicoscillatorortheunperturbedhydrogenatom,respectively.Methodsareconstantlybeingadvancedtocopewiththevarietyofperturbedsystemsthatmayrequireapproachesandmethodstobesolvedeffectively.Choosinganapproachfundamentallydeterminestheformoftheresult.Ananalyticalapproachyieldsanalgebraicexpression,whichcouldinvolveparameterdependencies,butrequiresgreateffortsinthecalculation.Acomputer-basednumericalapproach,whichautomaticallyyieldsaresult,lacksanypossibilitytorelatetheresultalgebraicallytoparametersortheinitialcondi-tions.Thisthesis,however,isconcernedwithahybridmethodofanalyticalandnumericalcalculations.Itusesadifferentialalgebra(DA)basednumericalapproximationofsolutionstoperturbedperi-odicsystems.TheDAframeworkwasdevelopedtoitscurrentextentbyMartinBerzet.al[3,4,5].Itallowsforthealgebraicstructuretobestoredinso-calledDAvectorswhichareusedastheelementsofthenumericalcalculation.Theapproachconsistsofthreemainsteps.First,theordinarydifferentialequation(ODE)asaresultofHamiltonequationsistransformedtocoordi-nateswhichmakethemsuitableforthethirdstep.TheODEisthenintegrated,inthesecondstep,tocreateatransfermap.DuetotheDAframeworkthismaprelatesthestatetotheinitialstateandODEinherentparametersalgebraically.Inthelaststep,thetransfermapistransformed1to'NormalForm'coordinates.InNormalFormcoordinatesthemaprepresentscircularmotioninphasespacewithonlyamplitudedependentfrequencies.Thesefrequenciesarethekeyquantitiestodeterminethebehaviorofperiodicsystems.ForthissteptheDANormalFormAlgorithm[6]isused,whichisanadvancementbyMartinBerzfromthearbitraryorderalgorithmthatwasbasedonanDA-LieapproachbyForest,Berz,andIrwin[13].ThewholemethodisimplementedinCOSYINFINITY[18],whichisalreadyequippedwiththeDAframeworkandtheDANormalFormAlgorithmdocumentedin[7,16,17].Thegreatadvantageofthismethodisthatitcombinesanalyticalandnumericalcalculationsandisnotlimitedtotheordertowhichthecalculationscanbedone.Thecomputer-basedDAcalculationallowsthemanipulationofpolynomialtransfermapsuptothepointaccuracyofthecom-puter.Thereforeitispreciseandeffectiveatthesametime.TointroducetheadvantageoftheDAbasedcalculation,theresultofafourth-orderRunge-Kutta(RK4)shallbepresentedintheoriginalandtheDAbasedIntheclassiccalculation1.1)thesolutionistracedfromtheinitialstatealongthevectoroftheODE.-6pq-111-1Figure1.1TheplotshowstheclassicRK4-trackingofthePendulum-ODE(eq.5.4)fromthe(-1,1)-initialstateuntilt=p=4.Allparametershadtobegivenvaluesandweresetto1.2IntheDAbasedversion1.2)thewholeinitialstateset~ziniinthevectorisrepresentedbythemapMiniintheformof~zini=Mini=(q;p)andthenintegratedwithrespecttotime.TheresultingtransfermapM(q;p)=~zfrelatestheinitialstateset~zinitoitsstate~zfincludingvectorinherentparameters.Thetwodimensionalstatespacein1.2representingqandpcoordinate,couldevenbeexpandedbyfurtherdimensionfortheinvolvedparameters.3210123321012-3-2-10123-2-10123pt=0t=0:5pqt=0t=2t=0t=1qt=0t=pFigure1.2Thepurplearearepresentsasectionoftheinitialstatesetinphasespace(q0;p0)=([1:3;1:3];[1:3;1:3]).ThegreenareadenotesthetransfermapM(q0;p0)oftheinitialstatesetinthePendulumvector(eq.5.4)attimet=0:5;1;2andp,respectively.TheDAbasedRK4methodwithastep-sizeofh=0:01wasusedfortheintegrationtoalgebraicallyconnectthewholeinitialstatesetuptoordermwiththestate.Thestatesetisnormallyextendedovermoredimensionswhichalsoinvolvestheparameter-space.Forthe2D-illustrationtheparametersaresetto1.31.2BasicconceptsThefollowingsectionintroducesessentialconceptswhichareusedinthisthesisandareindispens-abletounderstandingcertainstepsorargumentslateron.Firstofall,theissuesofapproximatingpotentiallylongrealnumbersinthepointrepresentationtomakethemsuitableforthecomputer-basedcalculationsarepresented.BeforegivingabriefintroductiontotheDAframework,theapproximationofanalyticalfunctionsusingTruncatedPolynomialSeriesisdis-cussed.Themainprocessofdiagonalizationwhichisimportantforthedecouplingofthelinearpartsofthetransfermapslateronisthenpresented.AstrongfocusisonthecalculationofthetwodimensionalcaseusingtheTwissparameters[12].Subsequently,thekeycharacteristicsofAction-AnglecoordinatesaresummarizedandrelatedtotheNormalFormcoordinates.Lastly,theFlowOperatoranditswayoftime-expandingobservablesareintroduced.1.2.1FloatingPointnumbersFloatingpointnumbers(F)areusedinsciencetoapproximaterealnumbers(R)totheircantpartuptoacertainprecision.Arealnumbercanpotentiallyrequireanamountofdigitstoberepresented,dependingonthebaseofthenumeralsystemused.Thepointrepresenta-tionusesthemost1digitsandscalesthembyanexponenttotheappropriaterange.Inmostcases,includingtheIEEE754standard[21]whichwillbediscussedinmoredetailbelow,aedbaseisusedandtheandtheexponentareadjustedtoapproximatethenumber:significandbaseexponent:(1.1)Incontrasttotheedpointnumberrepresentation,thepointcanwbetweenthedecimalplaces,duetoappropriateadjustmentoftheexponent.Therefore,thepointrepresentationhastheadvantageofarangebasedprecision.Whilethe'notation'ofpointnumbersusesabase10representation,computing,incontrasttothat,usesabase2representationduetothebinarysystem.1mostalldigitsallowedbythecurrentprecisionofthepointrepresentation.4Itmightnotbeobviousatglance,buttherearevariouswaystooperationsonpointnumbers,especiallyconcerningtherounding.Let;;andbetheoperationsonFcorrespondingto+;;and=inR.Thefollowingexampleshowstheevaluationof1x+22atx=4inRandFwith3sdecimaldigits(precisionp=3)inbase10,whileroundinghalfupaftereachcalculationinF:R:4x0+2!61=x1!16x22!136==ˇˇF3:4x02!61x1!0:167x2x2!0:279103(1.2)NotethatF3136=0:2781036=0:279103,whichshowsthemainproblemofoperationsonpointnumbers.Notonlydoestheerroroftheinitialapproximationoftherealnumbersoccur,butthoseerrorsalsotendtogrowduetooperations.Tokeepthemassmallaspossibleandtomakeresultscomparablebetweencomputationsondifferentmachines,theIEEE754standard[21]wasintroduced.Itthepointnumberrepresentationandarithmetic,whichensuresthatallmachinesproducethesameoutputforthesamepointoperations.Italsoanotheroperation,whichcanbeasourceofadditionalerrors-thebase-conversion.Whilethecomputercalculatesinbase2,theresultisgiveninabase10-expressionmostofthetime.TostatewhichdigitsoftheF-calculation-resultareactuallyant,thestudyofthepropa-gationofthoseerrorsiscriticallyimportant.Calculations,proofsandotherimportantinformationonarithmeticcanbefoundin[14].COSYINFINITY,thecomputationprogramusedinthisthesis,adoptsthedoubleprecisionrepresentation,whichisa64-bit(1-bitsign+11-bitexponent+52-bitsystem,yieldingaprecisionof˘14decimaldigits.Mostresultsofcalculationswithinthiswork,areaccurateupto1014,dependingonthecomplexityofthepreviouscalculationandtheassociatedroundingerrorpropagation.51.2.2TruncatedPolynomialExpansionSimilartothewaypointnumbersapproximateapotentiallylongrealnumbertoitsdigits,truncatedpolynomialscanapproximateaTaylorpolynomialexpansionofananalyticalfunctionfat~auptotheordermmax:~f~a+~h=¥ån=0~h~Ñ~xn~f(~x)i~x=~an!ˇmmaxåi=0~h~Ñ~xn~f(~x)i~x=~an!:(1.3)Considerthefollowingthreeanalyticalfunctionsfi2C¥(R):f1(x)=exp(x)f2(x)=2+sin(x)cos(x)+x33f3(x)=1+x+x22+x36:(1.4)TheapproximationofthosefunctionswithtruncatedTaylorexpansionsaroundx=0upto3rdordercanallberepresentedbythesamefunctionf3:Tf1(x)=3Tf2(x)=3Tf3(x)=1+x+x22+x36=f3(x):(1.5)Thefollowingoftheequivalenceclassesanditspropertiesisaccordingtothein[6,p.91].Thenotation'=m'indicatesthattheexpressionsonbothsidesareequivalentuptoorderm.Ingeneral,thismeansthattheequivalencerelationf=mgbetweentwofunctionsf;g2C¥(Rn),isgivenwhenf(~0)=g(~0)and¶kif(~x)~x=~0=¶kig(~x)~x=~0forall0inandallkm.Thisallowsfortheofanequivalenceclass[f]mthatrepresentsallelementsfofthevectorspaceofdifferentialfunctionsC¥(Rn)withnrealvariablesthathaveidenticalderivativesattheoriginuptoorderm.Thepointzeroischosenoutofconvenienceandwithoutlossofgenerality,meaningthatanyotherpointmaybeselectedaswell.SinceTf,thetruncatedTaylorexpansionoffisalwaysequivalenttofitselfuptoorderm,thissmalltheoremisapparent[f]m=[Tf]m(1.6)whereTfistheexpansionat0.Tfcanbeusedastheequivalenceclassrepresentative.Hence,relation1.5canbesummarizedto:f1;f22[f3]3=1+x+x22+x36.Allequivalenceclasses[]naredenotedasnDn[6]wherenrepresentsthenumberofvariables.TomakenDnadifferentialalgebraitrequirestheofcertainoperationswhichwillbeintroducedinthenextsection.61.2.3DifferentialAlgebraThefollowingbriefoverviewoftheDAandarithmeticlargelydrawsfrom[6],whereBerzsummarizestheDAframeworkanditstechniquesfromhisearlierwork[3,4,5];pleaserefertothosereferencesforfurtherinformation.mDndenotestheDAfornvariablesanddifferentiationuptomthorder.TointroducetheDAframework,Berzillustratesthesimplestcase1D1in[6].Similartocomplexnumberswhichhavearealandimaginarybasiswithi=(0;1)C,the1D1canberepresentedinaconstantanddifferentialbasiswiththedifferentialunitddef=(0;1)[6].Whilethesquarepropertyoftheimaginaryunitisknowntobei2=(0;1)C(0;1)C=(1;0)Cthesquareofthedifferentialunitdhasthespecialpropertyofvanishingwithd2=(0;1)(0;1)=(0;0)[6].Inthefollowingpartthe2D1shallbediscussed,wheretheoperationsthatformthealgebrawiththetuples(x0;x1;x2)canbeasfollows:(x0;x1;x2)+(y0;y1;y2)=(x0+y0;x1+y1;x2+y2)(1.7)c(x0;x1;x2)=(cx0;cx1;cx3)(1.8)(x0;x1;x2)(y0;y1;y2)=(x0y0;x0y1+x1y0;x0y2+x1y1+x2y0):(1.9)Similartoabovewecancalculated2in2D1:d2=(0;1;0)(0;1;0)=(0;0;1):(1.10)Therefore,anytuplecanberepresentedby(x0;x1;x2)=x0+x1d+x2d2:(1.11)Justlikein1D1,theoperationsallowtheofatotalorderthatiscompatiblewithitsalgebraicoperations[6].Fromtheorder,thenamefordbeingordifferentialbecomesapparent,sinceitissmallerthananyrealnumber0=(0;0;0)=d3<(0;0;1)=d2<(0;1;0)=d<(x0;0)=x0:(1.12)Infact,ditisingeneralsosmallthatthe(m1)thpowerofdinmDnvanisheswhichisalsocalledthenilpotentelementofmDn.Thethreeoperationsandthetuplesofrealnumbersformaring7algebra,butnotasincethereisnotamultiplicativeinversein2D1forevery(x0;x1;x2)22D1.Asamatteroffact,onlytuples(x0;x1;x2)withx06=0haveamultiplicativeinverse(y0;y1;y2)22D1with(x0;x1;x2)(y0;y1;y2)(1:9)=(x0y0;x0y1+x1y0;x0y2+x1y1+x2y0)!=(1;0;0))y0=1x0y1=x1x20y2=x21x30x2x20:(1.13)Introducingtheendomorphism(structure-preservingmapfrom2D1intoitself)calledderivation¶,makes(2D1;¶)adifferentialalgebra.¶:2D1!2D1(1.14)(x0;x1;x2)7!¶(x0;x1;x2)=(0;x1;2x2):(1.15)Theoperationsbehaveasfollowsunderthederivationmapforu1;u222D1:¶(u1+u2)=¶u1+¶u2(1.16)¶(cu1)=c¶u1(1.17)¶(u1u2)=(¶u1)u2+u1(¶u2):(1.18)Whileequation1.16and1.17arerathertrivial,equation1.18requiresasmallderivation:(¶u1)u2+u1(¶u2)=(0;x1;2x2)(y0;y1;y2)+(x0;x1;x2)(0;y1;2y2)=(0;x1y0;x1y1+2x2y0)+(0;x0y1;x02y2+x1y1)=(0;x0y1+x1y0;2(x0y2+x1y1+x2y0))=¶(u1u2):(1.19)Accordingto[6]theDAarithmeticcangenerallybeexpandedtotheequivalenceclasses[f]nwhicharedenotedbynDn,withdi=[xi]8inandthefollowingoperations:[f]n+[g]n=[f+g]n(1.20)c[f]n=[cf]n(1.21)[f]n[g]n=[fg]n(1.22)8whichthealgebraonnDnandareequivalenttotheoperationsonC¥(Rn)uptoorderm.Introducingtheendomorphismderivation¶:¶n[f]n=[xn¶nf]n(1.23)makesitadifferentialalgebra,sincethefollowingsmalltheoremisgenerallyalsosimilartothederivationinequation1.19¶n([f]n[g]n)=[f]n¶n[g]n+[g]n¶n[f]n:(1.24)Operationslikethedivisionareonlyifthedivisor[f]mhasanon-zeroconstantpartwhichisduetothefactthatnDnisonlyaringandnota(seeabove).Thecompositionoftwofunctionsormapsf;g2nDn[f([g]n)]n=[f(g)]n(1.25)isonlypossibleifghasnoconstantpart[2].Ingeneral[f]ncanbebyitstruncatedTaylorexpansionasfollows[6,2.42+43]:[f]n=ååni=1kinak1;:::kndk11::::dknn(1.26)withak1;:::;kn=1Õnj=1kj!¶åni=1kifÕnj=1¶xkjj:(1.27)Hence,ak1;:::;knrepresentstheTaylorexpansioncoefoffanddk11::::dknnmaybeusedasthebasisofaDAvector,whichstoresthecoefaccordingly.Inthiswayitispossibletoapproximateanyf2C¥(Rn)byatruncatedpolynomialserieswhichisthenrepresentedwithitsTaylorexpansioncoefak1;:::;knuptoorderminaDAvector.TheimplementationandoperationsoftheDAvectorsinCOSYINFINITYbasedontheframeworkaboveareexplainedin[18,6,5].91.2.4DiagonalizationThegoalofdiagonalizationistotransformamatrix‹Ltoitsdiagonalform.Thediagonalformhasentriesonlyonthediagonalandthereforerequires‹Ltobeaquadraticmatrix.Furthermore,noteveryquadraticnnmatrixcanbediagonalized,butonlymatricesthataresimilartoadiagonalmatrix‹L.Twomatrices‹Aand‹Baresimilarifandonlyif(following:iff)thereexistsasimilaritytransfor-mationmatrix‹Tanditsinverse‹T1,suchthat‹B=‹T‹A‹T1:Annnmatrix‹Thasaninverse‹T1with‹T1‹T=‹T‹T1=I,whereIistheidentityiffallcolumnsof‹Tarelinearlyindependent.Tothosematchinglinearlyindependentcomponent-vectors,theeigenvectorsandeigenvaluesof‹Larerelevant.Theeigenvaluesliandthecorre-spondingeigenvectors~viofaquadraticnnmatrix‹Laredeterminedbytheeigenvalueequation:‹L~vi=li~vi.Tosolvetheeigenvalueequation,onlythenontrivialsolutionsoftheeigenvalueprob-lem‹LliI~vi=~0areofinterest.Inthiscase,nontrivialsolutionsareany~vi6=~0.Therefore,solvingDet‹LlI=0(1.28)yieldsthenontrivialsolutionsoftheeigenvalueproblem.Thecharacteristicpolynomialp(l)ofannnmatrix‹Lisintroducedinthiscontext.Itisthedeterminantoftheeigenvalueproblemmatrixandisthereforerelatedtoitseigenvaluesliasfollows:p‹L(l)=Det‹LlI=nÕi=1(lli):(1.29)Theimportantpropertyofthecharacteristicpolynomialofamatrix‹Linthecontextofdiagonal-izationisthatithasthesamerootsandhasthereforethesameeigenvaluesliasthecorrespondingeigenvectors~vi,asthecharacteristicpolynomialsofallmatricessimilarto‹L.Thispropertycanbeshowninthisshortproof:li‹T1~vBi=‹T1li~vBi=‹T1‹B~vBi=‹T1‹T‹A‹T1~vBi=‹A‹T1~vBi(1.30)10whichshowsthat‹T1~vBi=~vAiwitheigenvalueli.‹T~v‹Ai=~v‹Biand‹Tisthetransformationmatrixbetweenthetwosimilarmatrices‹Aand‹B.Finally,annnmatrix‹Lwitheigenvaluesliisdiagonalizableiffallncorrespondingeigenvectors~v‹Liarelinearlyindependent.Thefollowingproofofthistheoremwillalsorevealthetransformationmatrix‹Tforthediagonalization:‹T1=~vL1;~vL2;:::;~vLnand‹LadiagonalmatrixwithentryLii=li.Now,L‹T1=‹A~vL1;~vL2;:::;~vLn=‹L~vL1;‹L~vL2;:::;‹L~vLn=l1~vL1;l2~vL2;:::;ln~vLn=‹T1‹L(1.31)iffalleigenvectors~vL1arelinearlyindependent,thenthereisamatrix‹T,whichisinverseto‹T1andthus‹T‹A‹T1=‹L:(1.32)Inthisthesis,22matricesareakeyelementsincematricesofhigherdimensionsoftheform2n2ncanbetoadiagonalblockmatrix,whichhas2x2-submatricesonitsdiagonal.Forthisreason,the2-dimensionalcaseshallbeinvestigatedinmoredetail.Thefollowingdiagonalizablereal22matrix‹Lisconsidered:‹L=0BBB@abcd1CCCA:(1.33)Accordingtoequation1.29thecharacteristicpolynomialisgivenbyDet0BBB@albcdl1CCCA=(al)(dl)cb=l2Tr‹Ll+Det‹L=0:(1.34)Theeigenvaluescanbeobtainedbysolvingthequadraticequationfromabove:l=Tr‹L2vuuuuutTr‹L24Det‹L|{z}d:(1.35)11Dependingonthediscriminantd,theeigenvaluesareeitherreal,complex,ordegenerate.Ford>0theeigenvaluesarereal,aswellastheircorrespondingeigenvectors.Ford=0theeigenvalueswillberealanddegeneratel=Tr(L)2.Theinterestingcaseisd<0wheretheeigenvaluesandthecorrespondingeigenvectorsarecomplexconjugatepairs.Itisusefultowritetheeigenvaluesinthereim-notation,wheremagnituderandphasemcanbederivedfromtherealandimaginarypartoftheeigenvaluepairasfollows:Rel=Tr‹L2Iml=vuutDet‹LTr‹L24(1.36)r=qRe(l)2+Im(l)2=rDet‹L(1.37)m=arccos Relr!=arccos0BB@Tr‹L2rDet‹L1CCA:(1.38)Sincethearccos-functionissymmetric,witharccos(x)=-arccos(x),thesignofmmustbeAcommon[6]isusingsign(b),whichresultsinm=sign(b)arccos0BB@Tr‹L2rDet‹L1CCA:(1.39)CourantandSnyderintroducedtheTwissparameters[12],whichareasetofparametersconcern-ing22matriceswithcomplexconjugateeigenvectors.Consideringthematrix‹Lfromequation1.33,theTwissparametersareasfollows[12]:a=ad2rsinm(1.40)b=brsinm(1.41)g=crsinm:(1.42)Notethatthesignofmusingbassuresthatbisalwayspositive.TheTwissparameters12arenotindependentofeachother,butsatisfybga2=4bc+(ad)24r2sin2m=4(adbc)+(a+d)24Im2(l)=4Det‹LTr2‹L4Det‹LTr2‹L=1:(1.43)Rewriting‹LintermsoftheTwissparametersyields‹L=r0BBB@cosm+asinmbsinmgsinmcosmasinm1CCCA:(1.44)Theeigenvectors~vtotheeigenvaluel=exp(im)arethereforegivenasfollows:0BBB@0BBB@cosm+asinmbsinmgsinmcosmasinm1CCCA0BBB@eim00eim1CCCA1CCCA0BBB@xy1CCCA=0BBB@001CCCA0BBB@x(ai)sinm+ybsinmxgsinm+y(ai)sinm1CCCA=0BBB@x(ai)+ybxg+y(ai)1CCCA=0BBB@001CCCA(1.45)whereexp(im)=cosmisinm.Equation1.45isforx=bandthecorrespondingy=ai,whichcomposethecomplexconjugateeigenvectorpair~v=0BBB@bai1CCCA:(1.46)Inthecaseofb=0andthereforeb=0,anequivalentcomplexconjugatedeigenvectorpaircanbederivedto~v=0BBB@aig1CCCA(1.47)13usingequation1.45withy=gandx=ai.Forthetrivialcasewhereb=0^g=0,theoriginalmatrixisalreadyindiagonalformandnotransformationisrequired.Ingeneral,theeigenvectorscanbescaledbyacomplexnumberk.However,topreservethecomplexconjugatepropertyoftheeigenvectorpairs,thevectorswillonlybescaledbyrealnumbersinthiswork.Thematrix‹Lcanthenbediagonalizedto‹Lwiththetransformationmatrix‹T1=k~v+;~vanditsinverse,whicharegivenindetailasfollows:‹L=r0BBB@eim00eim1CCCA=‹T‹L‹T1(1.48)‹T1=k0BBB@bba+iai1CCCA(1.49)‹T=1Det‹T10BBB@aibaib1CCCA(1.50)wherekisanadditionalscalingfactorwhichcanbeusedtoscalethematrix,suchthatthemagnitudeofthedeterminateofthetransformationmatrixis1.Thisscalingisessentialforthetransformationofnonlinearterms,sinceitensuresascaling-neutraltransformation.1.2.5Action-AnglecoordinatesAction-Anglevariablesareaspecialsetofphasespacecoordinates(J;q),whichamomen-tumJcalledactionthatisconstantanduniqueforeachphasespacecurve.Eachpointonthephasespacecurveisassociatedwiththecoordinateqcalledactionangle.AnyHamiltonianthatcanbecanonicallytransformedtothosecoordinatesiscalledintegrable.Inthecaseofaonedegreeoffreedomproblem,thenewHamiltonianKinthenewphasespacecoordinatesystemthenonly14-6-6-pqJqFigure1.3IllustrationofphasespacecoordinatetransformationtoAction-Anglecoordinates.dependsuponthenewmomentumJ1.3).Thus,theHamiltonequationsyieldJ=¶K¶q=0(1.51)q=¶K¶J=w(J)=const:(1.52)HavingKasafunctionofJaloneassuresthattheactionJisconstantfollowingtheHamiltonequation1.51foreachphasespacecurve.Hence,q=wfromequation1.52isalsoconstantandthereforeq(t)=q(0)+w(J)t.Action-anglecoordinatesaremainlyusedwhentheHamiltonianofthesystemisnotexplicitlytime-dependent,whichresultsintheconservationofH.Thegen-eralizedmomentum,whichresemblestheactionforeachoriginalgeneralizedcoordinatecanthenbeas:J(E0)=12pIp(E0;q)dq(1.53)wherethe2pisconventionaldependingonwhetherw(frequencyfornormperiod=2p)orn(frequencyfornormperiod=1)isused.Jishencethephasespaceareasweptoutinoneperiod=2p.Ingeneral,itdoesnotfollowthatifHand(p;q)satisfytheHamiltonequation,thenewphasespacecoordinates(J;q)andthenewHamiltonianKwilltoo.Thisisonlythecaseifthetransformationiscanonical,whichmeanstheremustbeageneratingfunctionofthesecondtypeF2(q;J)=S(q;J)ofoldcoordinateqandnewmomentumJ.ThisfunctionisthenasolutionoftheHamilton-Jacobiequation[1]K(J)=q;¶S(q;J)¶q(1.54)15whichyieldsthefollowingrelations:p=¶S¶qandq=¶S¶J:(1.55)Theexactderivationandfurtherinformationoncanonicaltransformationscanbefoundin[22,1].TheconceptofAction-Anglevariables(J;q)resemblesthebasicideaoftheNormalFormcoordinatest+;tin2Dwhicharelaterintroducedindetail.Bothsystemspossessaconstantofthephasespacecurveofthesystem:J˘r2=t+2+t2andthefrequencyisonlydependentonthatconstantofthecurvew(J)˘w(r2).1.2.6TheFlowOperatorSincethisworkdealswithcoupledorderdifferentialequations,itisusefultodiscusspossiblemethodstoinvestigatethetimeevolutionofacertainobservableO(~r;t)ofasystem.Itisassumedthatthetimederivatives~rofthesystem'scoordinatesaregivenbythefunction~f(~r;t).ThetimederivativeoftheobservableOistherefore:ddtO(~r;t)=¶O¶t+d~rdtdOd~r=¶t+~f~ÑO=L~fO(~r;t)(1.56)ApplyingtheFlowOperatorL~f=¶t+åni=1ri¶iisequivalenttotakingthetimederivative[6].L~fisalsoreferredtoasthederivativeoperator[19]andrepresentsthevectorof~f.Forthetimeevolution,theTaylorexpansionofOt0(~r;t)att0=0isconsidered[2]:O(~r;t)=¥ån=0tnn!dndtO(~r;0)=¥ån=0tnn!Ln~fO(~r;0)=exptL~fO(~r;0)(1.57)Foraone-dimensionalHamiltoniansystemH(q;p;t),thefunction~f(~r;t)canbederivedfromtheHamiltonequations:~f(~r;t)=~r=0BBB@qp1CCCA=0BBB@¶H¶p¶H¶q1CCCA(1.58)16TheFlowOperatorcanthenbewrittenas:L~f=¶t+~f~Ñ=¶t+q¶q+p¶p=¶t+¶H¶p¶q¶H¶q¶p(1.59)Thisworkonlydealswiththetimeevolutionofthetrivialobservable~r=Iinanautonomousandoriginpreservingsystem,whichyieldsthefollowingtermsrelevantfortheexpansion:L0~f~r=~rL1~f~r=~fL2~f~r=L1~f~f:(1.60)Thetimeexpansionatt0isthereforegivenasfollows:~rt0(t)=exp(tt0)L~ft0I(1.61)1.2.6.1ExampleofsymmetricallyperturbedHamiltonianIntheexampleofthesymmetricallyperturbedHamiltonian,thetimeevolutionshallbeexaminedusingtheFlowOperator.ThefollowingHamiltonianisconsidered:H=p22+q22+a4p2+q22(1.62)Forthetimeevolutionof~r,thevectorconsistingofthetimederivative~risneeded,whichcanbederivedfromtheHamiltonequations:~r=0BBB@qp1CCCA=0BBB@p1+ap2+q2q1+ap2+q21CCCA=~f(~r)(1.63)Thespecialtime-independentpropertyofHmakesitaconstantofthemotion.ButsinceH=r22+ar44=const.withr2=q2+p2,thismeansthatr2isalsoaconstantofthemotion.Therefore,~f(~r)canberewrittento~f(~r)=0BBB@qp1CCCA=1+ar20BBB@01101CCCA0BBB@pq1CCCA=0BBB@p1+ar2q1+ar21CCCA(1.64)17Fromequation1.60thetermsofthetimeevolution~rf(t)=exptL~fIaregivenasfollows:L0~f~r=0BBB@qp1CCCAL1~f~r=0BBB@pq1CCCA1+ar2(1.65)L2~f~r=q¶q+p¶p0BBB@pq1CCCA1+ar2(1.66)=0BBB@qp1CCCA1+ar22(1.67)L3~f~r=q¶q+p¶p0BBB@qp1CCCA1+ar22(1.68)=0BBB@pq1CCCA1+ar23(1.69)L4~f~r=q¶q+p¶p0BBB@pq1CCCA1+ar23(1.70)=0BBB@qp1CCCA1+ar24=L0~f~r1+ar24(1.71)18CollectingthetermsfortheTaylorexpansionaccordingtotheequation1.57yieldsthefollowingtimeevolution:~r(t)=0@L0~f+tL1~f+t2L2~f2+t3L3~f6+t4L4~f24+:::1A~r(1.72)=0BBB@qp1CCCA0B@1t21+ar222+t41+ar2424:::1CA(1.73)+0BBB@pq1CCCA0B@t1+ar2t31+ar236+:::1CA(1.74)=0BBB@qp1CCCAcos1+ar2t+0BBB@pq1CCCAsin1+ar2t(1.75)=0BBB@cos1+ar2tsin1+ar2tsin1+ar2tcos1+ar2t1CCCA0BBB@qp1CCCA(1.76)Thisresultisverysimilartothecommonlyknownsolutionofthesimpleharmonicoscillator.Bothrepresentacircularmotioninphasespacewiththeonlydifferencebeingthatthefrequencyistheradius/amplitudedependent.Inbeamphysics,thefrequencyofthelinearizedunperturbedsystemisreferredtoastunewhichiswandunityinthiscase.Thetuneshiftsareastheamplitudedependentfrequencychangesfromthetuneduetotheperturbation.Solutionsinthissetofcoordinates,which,inthiscase,coincideswiththeoriginalcoordinateswilllaterbeintroducedasNormalFormcoordinates,whichisonereasonforselectingthisexample.19CHAPTER2INTEGRATORSTherearevariouswaystoapproximatesolutionstoordinarydifferentialequations(ODE)numer-ically.Inthefollowingsection,threedifferentintegratorsshallbeintroducedthatmaybeusedtotrackthephasespacecurvefromtheinitialstatetothestate.Incontrasttocommoninte-gration,wheretheintegrationisdonealongonephase-spacecurve,theDAframeworkallowsthecomputationoftransfermapsateachstepoftheintegration;relatinganarbitraryinitialstatevector~ritothestatevectoratthatparticulartimeofthestepalgebraically~rf=M(~ri).Theinte-gratorisbasedonaTaylorexpansion,whichusestheFlowOperatorfrom1.2.6.ThecommonlyknownandusedRK4integrator,whichsimulatestheTaylor-expansionof4th-Orderisintroducedafterwards.Thelastintegratorusestheedpointtheoremandthecontractingpropertiesofanti-derivationoperationintheDAframeworktoapproximatethesolution.Allintegratorsworkfortimedependentaswellastime-independentsystems.2.1FlowIntegratorTheFlowIntegratoroperatesviaatime-wisestepbystepintegration.ThebasicprincipleofasinglestepisaTaylorexpansionintimeof~rn=In=(qn;pn)attnbyusingtheFlowOperator(1.2.6)andthenevaluating~rn(t=tn+h)=~rn+1(tn+1)[2].Afterwards,thecoordinatesof~rn+1arerewrittenintermsoftheoriginalcoordinates~r0=(q0;p0).TheapproachofusingtheFlowOperatorisknownfrompublicationssuchas[11],butsincethemultipleapplicationoftheFlowOperatorL~fcanbecomeveryextensiveforevenslightlycomplexfunctions,itisnotverypractical.However,thedifferentialalgebrasnDv+1withnpositionvariablesandonetimevariableimple-mentedinCOSYINFINITY,allowfortheautomaticdifferentiationwhichmakesthecalculationveryefTheFlowIntegratorisimplementedasfollows:M(n+1)(~r0;tn+1)=exphL~ftn=t0+nhIt0=0Mn(~r0)(2.1)20whereMnisthemapattimetn=nhandthereforeM0=~r0=I.Inthecaseofanotexplicitlytime-dependentsystem,where¶t~f=0iffollowsthat:exphL~ftn=t0+nhIt0=0=M18n(2.2)whereM1istheinitialwfromt=0tot=h,thereforeequation2.1canberewrittento:M(n+1)(~r0)=M1Mn1(~r0)=Mn+11(~r0)(2.3)Theordermtowhichtheprocessisdonecanbechosenarbitrarily.Thereore,itisapparentthatthetruncationerrorofastepis˘O(hm+1),whiletheglobaltruncationerroris˘O(hm)[19].Toillustratetheprocessofatwostepiteration,thefollowingexampleiscalculated,with~f=0BBB@qp1CCCA=0BBB@pt1CCCA:(2.4)Thetime-expansionatt0isderivedasfollows:0BBB@qp1CCCA=0BBB@qjt0pjt01CCCA+0BBB@qjt0pjt01CCCA(tt0)+0BBB@L~fqjt0L~fpjt01CCCA(tt0)22+0BBB@L2~fqjt0L2~fpjt01CCCA(tt0)36:::(2.5)=0BBB@qt0pt01CCCA+0BBB@pt0t01CCCA(tt0)+0BBB@t011CCCA(tt0)22+0BBB@101CCCA(tt0)36(2.6)Equation2.6showsthattheexpansionalreadyterminatesafterthe3rdorderint.Therefore,theentiretimeevolutioncanberepresentedbythisexpansionandthe2-stepapproach,whichisusuallydoneforahigherprecision,isnotneeded.However,forillustrationpurposesatwo-stepintegrationwithstepsizeh=1ispresented.Themapcanbealreadyterminatedbyusingequation2.6witht=2andt0=0,yieldingthefollowingmap:M2(q0;p0)=0BBB@q2p21CCCA=0BBB@q0+2p0+43p0+21CCCA(2.7)21Forthestepbystepintegration,thestepisachievedbyusingthesameequationasabove(2.6)fort0=0andt=tt0=h=1,yieldingthefollowingmap:M1(q0;p0)=0BBB@q1p11CCCA=0BBB@q0+p0+16p0+121CCCA(2.8)Thenextstepintheexpansionisperformedatt1=t0+h=1andagain,tt1=h=1:M2(q1;p1)=0BBB@q2p21CCCA=0BBB@q1+p1+1+16p1+1+121CCCA(2.9)M2(q0;p0)=0BBB@q0+p0+16+p0+12+1+16p0+12+1+121CCCA=0BBB@q0+2p0+43p0+21CCCA(2.10)Theresultinequation2.10agreeswiththeexpectedresultfromequation2.7.Notethatinequation2.10M2(q1;p1)=M1!2andthatM2(q0;p0)=M1!2M1(q0;p0)withM1!26=M1,duetotheexplicittimedependenceof~f.Sincethesystemisexplicitlytimedependent,theevaluationofeachstepisneeded.M21(q0;p0)=0BBB@q0+p0+16+p0+12+16p0+12+121CCCA6=0BBB@q0+2p0+43p0+21CCCA=M2(q0;p0)(2.11)222.2Fourth-OrderRunge-KuttaintegratorThefourth-orderRunge-Kuttamethod(RK4)simulatestheaccuracyoftheTaylorseriesmethodoforderm=4[19].Itisoneofthemostcommonlyusedsinceitisstable,quiteaccurate,andeasytoimplement.ItwasdevelopedbytheGermanmathematiciansCarlRungeandMartinKuttain1901.Itconsistsofastep-wiseintegrationoftheindependentvariablet.Givenasetofinitialconditions~rn=~r(ti),onestepoftheRunge-Kuttamethodyieldsanapproximationfor~rn+1=~r(tf=ti+h),wherehisthestep-sizeoftheintegration.Inthesimplestformofthemethod,thestep-sizewillbeconstantthroughouttheintegration.Moreadvancedintegratorsuseanadaptivestep-size[15].AseriesofRunge-Kuttastepsmakeitpossibletotracethetrajectory~rfromthesetofinitialconditionstothestate~rf(tf)instepsofh.Thisworkusesa4th-OrderRunge-Kuttamethod(RK4)withaconstantstepsizeh,where~rn+1attn+1=tn+hiscalculatedasfollows:~rn+1=~rn+h6~an+2~bn+2~cn+~dn(2.12)with~an=~f(~rn;tn)(2.13)~bn=~f~rn+h2~an;tn+h2(2.14)~cn=~f~rn+h2~bn;tn+h2(2.15)~dn=~f(~rn+h~cn;tn+h)(2.16)Thederivationofthecoef~an;~bn;~cnand~dncanbefoundin[8].DuetotheDAbasedimplementationthecalculationyieldsatransfermapthatrelatestheinitialstatetothestate.However,theordinarymethod(notDAbased)canonlybeusedforelement-by-elementtracking,yieldingonlyanumericalrelationbetweentheinitialandvalue.232.3FixedpointIntegratorTheedpointIntegratorturnsanordinarydifferentialequation(ODE)intoaedpointproblemwhichitsolvesthroughiteration.Hence,thefollowingshortintroductiontoedpointproblemsisgiven.AfunctionFhasaedpointx0ifF(x0)=x0isAssumingthatF(x)hasaedpointx0,thequestionarisesunderwhatconditionscantheedpointbeapproximatedthroughtheiterationxn+1=F(xn):(2.17)Itisapparentthattheedpointmustbeattractiveforthestartingvalueoftheiteration.There-foreF(x)mustbecontracting.AmapF:M!Monametricspace(M;d)withametricdiscontracting,if8x;y2Md(F(x);F(y))<kd(x;y)(2.18)wherekisarealnumber0k<1whichiscalledaLipschitzconstant.MthereforethespaceofcontractionofF.Furthermore,ifkd(xn;yn)>d(F(xn);F(yn))=d(xn+1;yn+1)n!¥!d(x0;x0)=0(2.19)thenFiscontractingonMwiththeedpointx0andtheiterationprocesswhichwillyieldtheedpoint.ConsideringtheexampleofF(x)=x2,oneedpointistriviallyx0=0andisattractivefortherangeofjxj<1.Theotheredpointisatx1=1,whichhasarangeofzeroandisthereforenotattractive.Ontheotherhand,G(x)=pxhasaedpointatx0=1,whichisattractiveforallx2R+,whichiseveryvalueofthedomainofGexceptfortheonlyotheredpointx1=0.Accordingto[2]theintegratorsolvestheedpointproblemforthefunction~R(~r(t);t)with~R(~r(t);t)=~r(0)+Zt0~f~r(t0);tdt0;(2.20)where~f=~rand~r(0)=I.Theedpointof~R(~r(t);t)isobviously~r(t),whichshallbeapproximatedbyedpointiteration(eq.2.17).Theintegration,whichisrepresentedbyananti-24differentiationoperation¶1intheDAframework,iscontractingwithrespecttothedepthnon-vanishingderivative)[6].Theedpointproblem~r(t)=I+Zt0~f~r(t0);t0dt0;(2.21)canthereforebeapproximated,orderbyorder,throughtheiteration:~rn+1(t)=I+Zt0~f~rn(t0);t0dt0;(2.22)with~r0(t)=~0.So,witheachstepoftheiterationthefollowingisvalidformbeingatleastn.~r(t)=m~rn(t)(2.23)AsanillustrativeexamplethesameODEintroducedpreviouslyfortheFlowIntegratorisapprox-imatedwiththeedpointmethod:r0(t)=I+Zt0~f~0;t0dt0=r(0)=0BBB@qp1CCCA(2.24)r1(t)=I+Zt0~fr0;t0dt0=0BBB@qp1CCCA+0BBB@ptt221CCCA(2.25)r2(t)=I+Zt0~fr2;t0dt0=0BBB@qp1CCCA+0BBB@ptt221CCCA+0BBB@t3601CCCA(2.26)and8n>2rn(t)=r2(t).Inthisexample,theiterationterminatesafterthe3rdorder.ThealgorithmcanbeimplementedinCOSYINFINITYinaveryefway,byconstrainingtheorderofeachsteptothenecessaryminimum,whichistheorderoftheiterationitself.Sotheseconditeration(eq.2.25)wouldberestrainedto2ndorderandsoon;thisavoidsunnecessarycalculationsofhigherordersineachstep,whichbecomedrasticallyevenmoretimeconsumingwitheachadditionalorder.252.4Errorvsstep-sizeforRK4integratorThefollowingsectioninvestigatestheerrorintheRK4integration.Ingeneral,theerrorsaredistinguishedinsingle-stepandmulti-steperrors,eachofwhichisdeterminedbythedifferenceoftheRK4-calculatedresulttotheexactsolution.FortheRK4,thesingle-steperrordependsonthestepsizehwithO(h5)[19].Themulti-steperror,whichisthetotalerrorfromallthesinglestepscombined,dependsonthestep-sizewithO(h4)[19].ThispropertycanbeshowninthefollowingexampleusingtheHamiltonequationsresultingfromthePendulumoscillation,whichisdiscussedindetailinsection5.1:0BBB@01101CCCA0BBB@p1q11CCCA+0BBB@aq313!a2q515!+a3q717!a4q919!+:::01CCCA(2.27)TheODEisintegratedusingasingle-RK4-stepfromt=0tot=h0.Theresultingmapyieldsthestartingpointforthefollowingtest:Toshowthatthemulti-steperror,whichisalsocalledtheglobaltruncationerror,isoforderO(h4),theintegrationtot=2h0isdoneinmultipleways:1;2;4;:::;32768stepsaredonewiththecorrespondingstep-sizesofh0;h02;h04;::::;h032768.Theresultofthesmalleststepsizeisconsideredtheexactresult.Hence,theerroristhedifferenceoftheresultfromtherespectivestep-sizetothe'exact'result.Theerrorisplottedagainstthenumberofstepsusedandshowninalog-logscalingin2.1.Alternatively,thereciprocalrelativestepsizecanbeused,whichisequivalenttotheusedstepsinthisexample.Thelinearcorrelationyieldsanh4-dependency,whichtheexpectedresult.Notethatforverysmallstep-sizestheconstanterrorthatoriginatesfromthepointnumberapproximations(seesec.1.2.1)becomesthedominantpartofthetotalerror.Accordingly,thestep-sizecanpotentiallybechosenineffectivelysmall,whereanincreaseinsteps,andthereforecomputationtime,doesnotresultinmorepreciseresults.Inthisexample,thecriticalstep-sizesisapproximatelyh0=1024ˇ104h0.TheothertestconcernsasinglestepoftheRK4method,wherebytheerrorequivalentlyasabove,issupposedtobeoftheorderO(h5)[19].Again,themapfort=h0isusedasthestartingpointfromwhichonestepofthedifferentstep-sizesh0=2;h0=4;h0=8;:::;h0=256isdone.2624524023523022522021521025202022242628210212normalizeddifferencetoexactintegration1=Stepsize1=x4Figure2.1ThegraphillustrateshowthetotalerrorofaDAbasedRK4-integrationdrops˘h4untilacertaincriticalstepsizeisreached.Fromthisstep-sizeonwards,morestepsdonotresultinmorepreciseresultssincethemaximumprecision,determinedbythepointcalculationerrors,isreached.Forthisexample,thecriticalstep-sizeisapproximately210h0ˇ104h0.Theerrorisestimatedbycalculatingthedifferenceoftheresultfromthestep-size2kh0totheresultfromthestep-size212h0,whichisconsideredthe'exact'solution.Thedifferenceisthennormalizedtotherespectiveresultofthe20h0-step-size-calculation.Thedifferentplots/colorsrepresentthecoefofdifferenttermsofthet=2h0mapofthePendulumODE(eq.5.4)aftertheintegrationfromthet=h0mapusingadifferentnumberofstepsaccordingtothestep-size.Forcomparisonthe'exact'resultiscalculatedby256;128;64;:::;2stepsofh0=512,respectively.Thedifferenceofone-step-resulttothe'exact'resultisplottedagainstthenumberofsteps.The5th-orderdependencebecomesapparentinthelinearcorrelationofthelog-logscaledgraph,whichisshownin2.2.ItshouldgowithoutsayingthattheDAvariableh0intheintegrationwasonlyintroducedforthepurposeofillustratingtheerrorbehavioroftheDAbasedRK4-integrator.Inthegeneralapplication,thevariableh0isreplacedbythestep-sizevalue.Therefore,2724023523022522021521025202122232425262728normalizeddifferencetoexactintegration1=Stepsize(2=x)5Figure2.2Thegraphillustrateshowthesingle-steperrorofaDAbasedRK4-integrationstepdrops˘h5withhbeingthestep-size.Theerrorisestimatedbycalculatingthedifferenceoftheresultfromthestep-size2kh0totheresultfromthestep-size29h0,whichisconsideredtheexactsolutioninthiscase.Thedifferenceisthennormalizedtotherespectiveresultofthe21h0-step-size-calculation.Thedifferentplotsrepresentthecoefofdifferenttermsofthet=h0+hmapofthePendulumODE(eq.5.4)aftertheintegrationfromthet=h0mapusingonestepofstep-sizeh.Theadditionalfactorof25intheerrorto1/step-sizeratiooriginatesfromnormalizationwithrespecttothe21h0-step-size-calculation.thegeneratedtransfermapsdonotrepresentthestateafteranarbitrarytimeh0,butatimesuchast=1.Thegeneralizationofaddingastep-sizevariableisnotneededinmostcasesandwouldthereforeonlyincreasethetotalorderofthesingleterms.Sincethecalculationisordertruncated,thegeneralresultwouldbelessprecisethanthedirectcalculationswithavalue.Consideringanordertruncationof5,termslikeq2a2t2wouldnotberepresentedinthegeneralsolution,duetothetotalorderof6oftheterm.Butforastep-sizelike1,thetermisrepresentedtogetherwiththeotherq2a2-terms.282.5IntegratorcomparisonToinvestigatehowthevariousDAbasedintegratorscompare,theintegrationofthePendulumoscillationODE(eq.5.26)isinvestigated.Allintegratorskeptthestep-sizet=havariable.TheODEwasintegratedinonestepfromt=0tot.Theresultingtransfermapsfortheqandp-componentaregivenintable2.1and2.2,respectively.Theordertruncationwassetto11.BoththeedpointIntegratoraswellastheFlowIntegratoryieldthesameresultuptothesetordertruncation.Thisisnotverysurprisingsincebothofthemapproximatetheexactsolutionorderbyorder,whichresultsinbothyieldingthesameresult,namely,theexactsolution(withpointaccuracy)uptoorder11.TheRK4ontheotherhandsimulatestheTaylorexpansionupto4th-order,thereforealltermswithtoforder4orlowercoincidewiththeFloedpointintegration.Fromthetheoryaboveitisknown,thattheerrorofasingleRK4-stepisoforderO(h5).Thiscanbeobservedinthetables(2.1and2.2)aswell.Termsoftwithorder5orhigherareeitherzeroordifferenttotheedpoint/Flowintegration,justasexpected.29Table2.1Thetablerepresentstheq-componentofthetransfermapoftheinitialstatefromt=0totinthePendulumvectorfromequation5.26uptothe11thorder.ThetransfermapwascalculatedwithonestepviathethreedifferentDAbaseintegrators.ThecolumnswithboldordernumbersindicatetermsofO(t5),whicharetheerroraffectedtermsoftheRK4integrator,whichsupportsthetheorydiscussedinsection2.4.Furthermore,theresultsofedpointintegrationandFlowintegrationareidentical,whichalsoagreeswiththetheory.OrderFactorRK4(t)Fixed-P(t)Flow(t)1q1,00000E+001,00000E+001,00000E+002pt1,00000E+001,00000E+001,00000E+003qt2-5,00000E-01-5,00000E-01-5,00000E-014pt3-1,66667E-01-1,66667E-01-1,66667E-015qt44,16667E-024,16667E-024,16667E-026q3t2a8,33333E-028,33333E-028,33333E-026pt58,33333E-038,33333E-037q2pt3a8,33333E-028,33333E-028,33333E-027qt6-1,38889E-03-1,38889E-038q3t4a-2,77778E-02-2,77778E-02-2,77778E-028qp2t4a4,16667E-024,16667E-024,16667E-028pt7-1,98413E-04-1,98413E-049q5t2a2-4,16667E-03-4,16667E-03-4,16667E-039q2pt5a-2,08333E-02-3,33333E-02-3,33333E-029p3t5a6,94444E-038,33333E-038,33333E-039qt82,48016E-052,48016E-0510q4pt3a2-6,94444E-03-6,94444E-03-6,94444E-0310q3t6a5,20833E-035,78704E-035,78704E-0310qp2t6a-5,20833E-03-1,52778E-02-1,52778E-0210pt92,75573E-062,75573E-0611q5t4a25,55556E-035,55556E-035,55556E-0311q3p2t4a2-6,94444E-03-6,94444E-03-6,94444E-0311q2pt7a2,60417E-036,84524E-036,84524E-0311p3t7a-2,18254E-03-2,18254E-0311qt10-2,75573E-07-2,75573E-0730Table2.2Thetableshowsthep-componentequivalenttoTable2.1.OrderFactorRK4(t)Fixed-P(t)Flow(t)1p1,00000E+001,00000E+001,00000E+002qt-1,00000E+00-1,00000E+00-1,00000E+003pt2-5,00000E-01-5,00000E-01-5,00000E-014qt31,66667E-011,66667E-011,66667E-015q3ta1,66667E-011,66667E-011,66667E-015pt44,16667E-024,16667E-024,16667E-026q2pt2a2,50000E-012,50000E-012,50000E-016qt5-8,33333E-03-8,33333E-037q3t3a-1,11111E-01-1,11111E-01-1,11111E-017qp2t3a1,66667E-011,66667E-011,66667E-017pt6-1,38889E-03-1,38889E-038q5ta2-8,33333E-03-8,33333E-03-8,33333E-038q2pt4a-1,66667E-01-1,66667E-01-1,66667E-018p3t4a4,16667E-024,16667E-024,16667E-028qt71,98413E-041,98413E-049q4pt2a2-2,08333E-02-2,08333E-02-2,08333E-029q3t5a3,12500E-023,47222E-023,47222E-029qp2t5a-1,04167E-01-9,16667E-02-9,16667E-029pt82,48016E-052,48016E-0510q5t3a22,22222E-022,22222E-022,22222E-0210q3p2t3a2-2,77778E-02-2,77778E-02-2,77778E-0210q2pt6a4,68750E-024,79167E-024,79167E-0210p3t6a-2,25694E-02-1,52778E-02-1,52778E-0210qt9-2,75573E-06-2,75573E-0611q7ta31,98413E-041,98413E-041,98413E-0411q4pt4a25,55556E-025,55556E-025,55556E-0211q2p3t4a2-2,08333E-02-2,08333E-02-2,08333E-0211q3t7a-4,34028E-03-6,87831E-03-6,87831E-0311qp2t7a2,60417E-022,02381E-022,02381E-0211pt10-2,75573E-07-2,75573E-0731CHAPTER3THENORMALFORMALGORITHMInthepreviouschapter,thedifferentDAbasedintegratorswerediscussed,whichyieldatransfermap,thatalgebraicallyconnectstheinitialstatetothestateattimet:~zf=M~zi;~d,where~drepresentspossibleparameterdependencies.ConsideringarepetitiveHamiltoniansystem,wherethecomponentsoftheresultingmapareinphase-spacecoordinates,theNormalFormAlgorithmprovidesanonlinearchangeofvariablesforthegivenmapM,whichalltermsuptoanarbitraryordermmax.Inthetransformedvariables,thetransfermapwillrepresentcircularmotionwithonlyamplitudedependentfrequencies[6].ThosefrequenciesarethekeyquantityofeveryperiodicsystemandinmostcasesmaketheexplicittrajectoryThisformofthemapiscalledthe'NormalFormofM'orMNF,intheshorthandnotation.ThefollowingintroductiontotheDANormalFormAlgorithmlargelydrawsfrom[6]andiscom-plementedwithanexplicitcalculationforasymplecticsystemwithn=1upto3rdorder.Theischosen,sinceonlyHamiltoniansystemsarediscussedinthisthesis.Symplecticmapsarecanonical[6]andthereforepreservetheHamiltonianform(see1.2.5).3.1TheDANormalFormAlgorithmTheDANormalFormAlgorithmusestheDAframeworkimplementedinCOSYINFINITYtoexpresstheNormalFormofthemapMinalgebraicrelationtotheinitialstateusingasequenceoforder-by-ordercoordinatetransformationsAmappliedtothemapMinthefollowingway:AmMA1m(3.1)Inthegeneralform,themapM=C0+L+åmUmconsistsofaconstantpartC0,alinearpartLandthenonlinearpartsUmoforderm.Mis2n-dimensional,withnposition/momentumentry32pairsMj.Forn=1themapcanbeexplicitlywrittenasM(x;p)=0BBB@M+M1CCCA=0BBB@x0p01CCCA|{z}C0+0BBB@(xjx)(xjp)(pjx)(pjp)1CCCA0BBB@xp1CCCA|{z}L+0BBB@U+2(2;0)U2(2;0)1CCCAx2+0BBB@U+2(1;1)U2(1;1)1CCCAxp+0BBB@U+2(0;2)U2(0;2)1CCCAp2|{z}U2+0BBB@U+3(3;0)U3(3;0)1CCCAx3+0BBB@U+3(2;1)U3(2;1)1CCCAx2p+:::|{z}U3(3.2)InthestepoftheNormalFormAlgorithm,theparameter-dependentedpoint~zFix;~distranslatedtotheorigin,soM~0;~d=0tomakethemaporiginpreserving.Thisedpointrep-resentsthereferenceorbitinphasespace,whichcanoftenbeobtainedbysolvingtheunperturbedODEoftheproblem.InmostcasestheconstantpartC0pointstothereferenceorbit.Therefore,itcanberemoved,whichyieldsM0=L+åmUm.Following[6],theedpointproblemM~zFix;~d=~zFix;~d(3.3)hastobesolvedforthenontrivialcase,whichisdonebythefollowingequation:~zFix;~d=(MI)1~0;~d:(3.4)From1.2.3itisknownthatpolynomialswithnon-zeroconstantterms,liketheidentity,haveaninversewhichcanbecalculateduptoordermmaxusingequation3.30.333.1.1DiagonalizationThediagonalizationisimportantforthecomputation,sinceitdecouplesthe2nphasespaceintonsubspaces,whichcanthenbetreatedindependently.Thematrix‹LisassociatedwiththelinearpartofM.Itistransformedintoadiagonal-block-matrix,whichhas2x2-submatricesonitsdiagonal:0BBBBBBBBBBBB@‹L1...‹Lj...‹Ln1CCCCCCCCCCCCAwith‹Lj=0BBB@ajbjcjdj1CCCA(3.5)Consequently,themapcannowbetreatedinthetwo-componentsubspace.Moreinformationontheblockmatrixcreationcanbefoundin[6].3.1.1.1DiagonalizationtransformationofthelinearpartIntheordercorrection,thelinearpartofthemapLmustbediagonalized.Thus,itisassumedthat‹Lisdiagonalizableandhasndistinctcomplexconjugateeigenvaluepairslj.Furthermore,itisrequiredthatnoeigenvalueisunityandtheproductofalleigenvaluesispositive,whichisgenerallytrueforarepetitivesystemundernormalconditions[6].Theordertransformationmatrix‹A11=~v1;¯~v1;:::;~vj;¯~vj;:::;~vn;¯~vnconsistsofthencomplexconjugateeigenvectorpairs~vj;¯~vjof‹Lresultinginthediagonalizedlinearmatrix‹Rwithcomponents:‹Rj=A1;j‹LjA11;j=0BBB@l+j00lj1CCCA=0BBB@eimj00eimj1CCCA(3.6)ItisessentialforthetransformationofthehigherordertermsthatjDet‹Aj=1tokeepthetransformationscaling-neutral.34Forthen=1theeigenvaluescanbederivedfromtheeigenvalueequation1.28whichyieldsthefollowingresultsfortheeigenvaluesandtheirphase:l=(xjx)+(pjp)2r((xjx)+(pjp))24(xjx)(pjp)+(pjx)(xjp)=reim(3.7)m=sign((xjp))arccos (xjx)+(pjp)2p(xjx)(pjp)(pjx)(xjp)!(3.8)where(r=1)^(m2R),sinceintheexampleasymplecticsystemisconsidered.Theeigen-vectorsof‹LcanthenbeexpressedintheTwissparametersasfollows:~v=0BBB@bai1CCCA(3.9)a=(xjx)(pjp)2rsinmb=(xjp)rsinmg=(pjx)rsinm(3.10)Tofollowtheexampleinthecaseofb=0,onecaneitherusetheeigenvectorsdescribedinequation1.47,whichareintermsofgandaandfollowthediagonalizationstepswiththoseoronecantransformthecurrentmaptoadifferentform,wherethenewb0=g,g0=1=ganda0=0.ThismapcanbegeneratedbyusingthetransformationA0with:‹A10=0BBB@1=ga0g1CCCA‹A0=0BBB@ga01=g1CCCA(3.11)A0LA10=0BBB@cosmgsinmsinmgcosm1CCCA(3.12)Thehigherordertermshavetobetransformedaccordingly(see3.1.1.2).Whetherwiththeoriginalmaporthetransformedmap,theeigenvectorsofthelinearparthaveamagnitudeandphase-freedomtothem.Inthiscalculationthephase-freedomisusedtomakesurethatthetransformationmatrixconsistofacomplexconjugateeigenvectorpair.Thisassurescertaincomplexconjugatepropertiesofthetransformedmap.Sincethetransformationofnonlinearterms35isalsorelevantinthisprocess,itisnecessaryforthetransformationmatrixtosatisfyjDet(A)j=1.The1st-ordertransformationmatrix‹A11=~v+;~vanditsinverse‹A1(see1.2.4)aregivenasfollows:‹A11=0BBB@(xjs+1)(xjs1)(xjs+1)(pjs1)1CCCA=1p2b0BBB@bba+iai1CCCA(3.13)Notethescalingfactorof1=jdet‹A11j,thatassuresascalingneutraltransformation.‹A1=1p2b0BBB@1+iaib1iaib1CCCA=0BBB@(s+1jx)(s+1jp)(s1jx)(s1jp)1CCCA(3.14)Forthelinearpartthetransformationyields:‹R=‹A1‹L‹A11=0BBB@l+00l1CCCA=0BBB@eim00eim1CCCA(3.15)3.1.1.2Diagonalizationtransformationofthenon-linearpartsThelineartransformationforthediagonalizationofthelinearpartalsotransformsthenonlinearpartsUmfromposition/momentumbasisx1~ex1;p1~ep1;:::;xj~exj;pj~epj;:::tothecomplexeigen-vectorbasisof‹L:s+1~v1;s1¯~v1;:::;s+j~vj;sj¯~vj;:::,asfollows:0BBB@xjpj1CCCA=A1:j0BBB@s+jsj1CCCA=0BBB@(xjjs+j)s+j+(xjjsj)sj(pjjs+j)s+j+(pjjsj)sj1CCCA(3.16)Themapaftertheordertransformationintotheeigenvectorbasiscanthereforebewrittenasfollows:M1=R+åmSm,whereSmarethetransformednonlinearparts,whichnowdependon~s+and~s,theeigenvectorcoefinsteadof~xand~p.Amoreexplicitformofthejth-36componentwithasimilarnotationasintroducedin[6,7.63]lookslikethis:Sm;j=åjj~k++~kjj1=mSm;jj~k+;~knÕl=1s+lk+lslkl(3.17)wherekjrepresentsthepositiveintegerexponentofsjsummarizedin~kandjj~kjj1:=åjkljistheL1-Norm(alsoknownasManhattanNorm),whichassuresthatonlypolynomial-termsofordermareconsidered.Sm;jj~k+;~kistheTaylorexpansioncoefwithrespecttoÕnl=1s+lk+lslkl.So,Mj~s+;~s=rjeimjsj+mmaxåm=2Sm;j(3.18)=rjeimjsj+mmaxåm=2åjj~k++~kjj1=mSm;jj~k+;~knÕl=1s+lk+lslkl(3.19)Alsonote,sinces+landslarecomplexconjugatepairs,thats+l=slandtherefore:Mj(~s+;~s)=rjeimjsj+mmaxåm=2åjj~k++~kjj1=mSm;jj~k+;~knÕl=1s+lk+lslkl=Mj~s+;~swithSm;jj~k+;~k=Sm;jj~k;~k+(3.20)Forn=1theexplicittransformationofthenonlinearpartsisdoneasfollows:Theindividualtrans-formationofsinglepartsofthemapisonlypossibleinthecaseofalineartransformation,whichisgivenforthediagonalization.Therefore,thetransformationofthetermUm(k+;k)xk+pkwithk++k=mcanbeshowningeneral:FirstA11transformsthenewcomplexconjugatevari-ables(s+;s)intotheoldvariables(x;p)tomakethemsuitableforUm(k+;k)xk+pk,whichisafunctionofxandp.1p2b0BBB@bba+iai1CCCA0BBB@s+s1CCCA=s2b0BBB@Rebs+Re(ai)s1CCCA(3.21)Theresultin(x;p)-coordinates,whichisrealsince(x;p)arereal,cannowbeinsertedintothenonlinearpartUm(k+;k)xk+pk:Um(k+;k)xk+pkA11=2bm2Um(k+;k)Rebs+k+Re(ai)sk(3.22)37Inthelastpartofthetransformation,thecurrentresultin(x;p)-coordinatesistransformedbacktos+;s-coordinateswithA1Um(k+;k)xk+pkA11=122bm+12Rebs+k+Re(ai)sk0BBB@ia1biia1bi1CCCAUm(k+;k)(3.23)=122bm+12Rebs+k+Re(ai)sk(3.24)0BBB@0BBB@ia1iai1CCCAU+m(k+;k)+0BBB@bibi1CCCAUm(k+;k)1CCCA(3.25)Theresultofthetransformationisavectorwithcomplexconjugateentries.Sincealltransformedtermscanjustbeaddedduetothelinearityofthetransformation,theresultM1willhavecomplexconjugateentriesM=0BBB@M+M1CCCA=0BBB@MM+1CCCA(3.26)asalreadysuggestedinequation3.20.Afterthetransformation,alltermswillbeexpandedandsummarizedwithrespecttothenewcoordinatess+k+skgivingthenewTaylorexpansioncoefSm(k+;k)wherem=k++k.Thetwocomponents()ofthenewmapM1looklikethis:M1=eims+mmaxåm=2åk++k=mSm(k+;k)(s+)k+(s)k(3.27)withS2=S2(2;0)(s+)2+S2(1;1)s+s+S2(0;2)(s)2(3.28)=0BBB@(s+js+s+)(sjs+s+)1CCCA(s+)2+0BBB@(s+js+s)(sjs+s)1CCCAs+s+0BBB@(s+jss)(sjss)1CCCA(s)2(3.29)383.1.2Thenon-lineartransformationThediagonalizationisfollowedbythenonlineartransformations,whicharethekeytransforma-tionsofthealgorithm.Thetransformationsareconductedorderbyorderstartingwiththe2nd.Theprocessisequivalentforeachorderandisthereforedescribedingeneralforthemth-ordertransformationdrawinglargelyfrom[6].ThemapMm1,whichisalreadyuptoorderm1,istransformedasfollows:AmMm1A1m,whereAm=I+TmandA1m[6,7.60+61]:A1m=ITm+T2mT3m+:::(3.30)Sinceonlythemth-orderofMm1isrelevantforthedeterminationofthemth-ordertransforma-tion,allhigherordertermscanbeignored.Thetransformationyieldsthefollowingtermsofmthorder[6,cf7.62]:AmMm1A1m=m(I+Tm)(R+S)ITm+˘˘˘˘˘˘˘T2mT3m+:::(3.31)=m(I+Tm)(R(ITm)+S(IˆˆTm))(3.32)=m(I+Tm)(RRTm+S)(3.33)=mRRTm+S+Tm(R((((((RTm+S)(3.34)=mR+S<m+Sm+S>m+[Tm;R](3.35)Inequation3.31,alltermsofTnmwithn>1areirrelevant,sincetheyareatleastoforderm2.Inequation3.32,thetermSTmwascanceledforthesamereason-theresultingexpressionisatleastoforderm+1,justlikethetermTmSinequation3.34.Inthatsameequation,thetermTmRTmisirrelevant,becauseitisatleastoforderm2.ThetermS>minequation3.35denotesonlytermsoforder>mandthereforeisalsocanceled.ThegoalistoTm,suchthatthecommutatorCm=[Tm;R]=Sm,toeliminatealltermsoforderm.Thisisingeneralnotalwayspossibleandinthenextsteps,itwillbecomemoreapparentwhichtermsofSmcannotbeeliminated.SimilartoSm,thejth-componentofTmcanbewrittenas:Tm;j=åjj~k++~kjj1=mTm;jj~k+;~knÕl=1s+lk+lslkl(3.36)39Hence,thejth-componentofthetwopartsofthecommutatorcanbewrittenasfollows:(TmR)j=åjj~k++~kjj1=mTm;jj~k+;~knÕl=1rle+imls+lk+lrleimlslkl(3.37)=åjj~k++~kjj1=mTm;jj~k+;~kei~m~k+~knÕl=1rk+l+klls+lk+lslkl(3.38)(RTm)j=rjeimjTm;j(3.39)Combiningbothto(TmRRTm)j=Cm;jresultsin[6,cf7.64]:Cm;j=åjj~k++~kjj1=mCm;jj~k+;~knÕl=1s+lk+lslkl(3.40)Cm;jj~k+;~k=Tm;jj~k+;~k ei~m~k+~k nÕl=1rk+l+kll!rjeimj!(3.41)Itfollowsimmediately,thatthetermassociatedwithSm;jj~k+;~kcanonlybeeliminated,ifthefactorCm;jj~k+;~kisnonzero,inwhichcaseTm;jj~k+;~kisgivenby[6,cf7.72]:Tm;jj~k+;~k=Sm;jj~k+;~k ei~m~k+~k Õnl=1rk+l+kll!rjeimj!(3.42)Cm;jj~k+;~kgiveninequation3.41iszerounderthefollowingcondition[6]:ei~m~k+~k nÕl=1rk+l+kll!=rjeimj(3.43)Fornon-symplecticsystems,thefurtherapproachcanbefoundin[6].Forsymplecticsystems,whererj=1^mj2R8j,theconditioninequation3.43toei~m~k+~k=eimj(3.44)andevenfurtherto[6,cf7.65]:mj(k+jkj1)+ål6=jml~k+~k=0(mod2p)(3.45)h~m;~k+~k~eji=0(mod2p)(3.46)40Inthetrivialcase,theconditionismetfor:k+jkj=1andk+l=kl8l6=j(3.47)whichyieldstermsthatareresponsiblefortheamplitude-dependenttuneshiftsandisthereforetheheartoftheNormalFormAlgorithm.Thisconditionispurelymathematical.Allothernon-trivialsolutionstoequation3.46areofaphysicalnature,whichmeansthat,purelymathematicallyitisquiteimpossibletoalinearcombinationofintegermultiplesoftheeigenvaluesmlsothattheresultisexactlymj(mod2p)evenforordersm˛10.Fromacomputational/physicalpointofviewitissufforthelinearcombinationtobecloseenoughtomj(mod2p)forTm;jj~k+;~kto'blowup'.Thiscaserepresentshigherorderresonances,whichareofaphysicalnature.Consideringtheconditionitbecomesapparent,thattheonlytermsthataremathematicallyimpossibletoremoveare:S+m;jj~k+~ej;~k(s+j)Õnl=1s+lslklSm;jj~k;~k+~ej(sj)Õnl=1s+lslklform=jj2~k+~ejjj1(3.48)whichonlyoccurforoddorders.ItisimportanttorecognizethatthenonlineartermsS>mwithahigherorderthantheorderofthecurrentcoordinatetransformationmareverylikelytochangeduetothemth-ordercoordinatetransformation.Thefollowingexampleofthenon-linear2ndordertransformationforn=1willillustratehowhigherorderslikethe3rdorderareaffected.InordertoeliminateallS2-termswiththe2ndordertransformation,T2inthetransformationA2=I+T2hastobe(T2jk+;k)=(S2jk+;k)eim(k+k)eim(3.49)accordingtoequation3.42.Tostudyhowthetransformationaffectsthehigherorders,the41secondordertransformationisconsideredupto3rd-order.M2=3A2M1A12(3.50)=3(I+T2)(R+S2+S3)IT2+T22:::(3.51)=3(I+T2)(R(IT2)+S2(IT2)+S3)(3.52)=3(I+T2)(RRT2+S2+S2!3+S3)(3.53)=3RRT2+S2+S2!3+S3(3.54)+T2R(((((((((((((RT2+S2+S2!3+S3(3.55)=3RRT2+S2+S2!3+S3+T2R(3.56)=3R+S2+[T2R]|{z}=0+S3+S2!3|{z}S3;new(3.57)Allthecrossed-outtermsinequation3.51and3.55representpartsthatwon'tcontributetotheresultuptoorder3,eachofthemisatleastoforderm+1.Asaresult,thereisanew3rd-orderpart,whichconsistsoftheunchangedoldS3andanewS2!3-term.Inthefollowingcalculation,thenewtermisexaminedfurther.S2!3=3S2(IT2)S2=3S2(2;0)s+T+22+S2(0;2)sT22+S2(1;1)s+T+2sT23S2=3S2(2;0)s+2+S2(1;1)s+s+S2(0;2)s2S2+((((((((((((((((((((((((S2(2;0)T+22+S2(1;1)T+2T2+S2(0;2)T222S2(2;0)T+2s+S2(1;1)T+2s+T2s+2S2(0;2)T2s=32S2(2;0)T+2s+S2(1;1)T+2s+T2s+2S2(0;2)T2s(3.58)wheretheT22termscanbeneglectedbecausetheyareoforder4.NotethattheT2-termsalsodependonS2asequation3.49shows.Itbecomesapparentthatthetransformationhasa42effectonthehigherordertermsandthereforeitisnecessarytodetermineanupperboundfortheordertowhichthenormalformiscalculatedbeforehand.Forthenextstep,the3rd-ordercorrection,thesamecalculationsaredone.Assumingthatthetruncationordermmax=3,itwillnotbenecessarytocalculatethetransformationexplicitly,becauseitisknownfromequation3.48,whichtermswillcanceloutandthechangestothehigherordersareirrelevant.OnlythetermsS+3;new(2;1)andS3;new(1;2)cannotbecanceled,duetotheconditioninequation3.47.Thereforetheremainingpartsofequation3.58aregivenasfollows:S+2!3(2;1)=2S+2(2;0)T+2(1;1)S+2(1;1)T+2(2;0)+T2(1;1)2S+2(0;2)T2(2;0)=2S+2(2;0)S+2(1;1)1eim+S+2(1;1)S+2(2;0)e2imeim+S+2(1;1)S2(1;1)1eim+2S+2(0;2)S2(2;0)e2imeim(3.59)S2!3(1;2)=2S2(2;0)T+2(0;2)S2(1;1)T+2(1;1)+T2(0;2)2S2(0;2)T2(1;1)=2S2(2;0)S+2(0;2)e2imeim+S2(1;1)S+2(1;1)1eim+S2(1;1)S2(0;2)e2imeim+2S2(0;2)S2(1;1)1eim(3.60)Notethatcomplexconjugaterelationstillholdswith:S+2!3(2;1)=S2!3(1;2).Unfortunately,theexpressionofS+2!3(2;1)intermsofU2isalreadyverylongandthereforenotgiven.Thisshowshowthecomputer-basedcalculationusingtheDAframeworkinCOSYINFINITYisessential.TheoriginalS3-term,whichistheentireS3;new-termforU2=0,canbederivedinashortformulaintermsofU3:S+3(2;1)=14g(3U3(0;3)+U+3(1;2))+b(3U+3(3;0)+U3(2;1))2a(U+3(2;1)+U3(1;2))3i4U+3(0;3)g2U3(3;0)b2+a(g(U3(0;3)U+3(1;2))+b(U3(2;1)U+3(3;0)))i4(U+3(2;1)+U3(1;2))(bg+a2)=S3(1;2)(3.61)Aspreviouslynoted,alltermssurvivingafterthe3rd-ordertransformationarethefollowing:M3=0BBB@M+M1CCCA=30BBB@s+e+im+S+3;new(2;1)s+sseim+S3;new(1;2)s+s1CCCA(3.62)433.1.3TransformationtoNormalFormcoordinatesInthelaststep,themapistransformedintoNormalForm.Afterthemmax-ordertransformation,thejth-componentofMhasthefollowingform[6,7.66].0BBB@M+jMj1CCCA=0BBBB@s+je+im+åmmaxm=jj2~k+~ejjj1S+m;jj~k+~ej;~kÕnl=1s+lslklsjeim+åmmaxm=jj2~k+~ejjj1Sm;jj~k;~k+~ejÕnl=1s+lslkl1CCCCA(3.63)=0BBB@s+jfjs+1s1;:::;s+lsl;:::;s+nsnsj¯fjs+1s1;:::;s+lsl;:::;s+nsn1CCCA(3.64)Thesimtofjanditscomplexconjugate¯fjaboveispossibleduetothecomplexconju-gaterelationshipbetweenM+jandMjshowninequation3.20.Rewritingfj=ajeifjyieldsMj=sjajeifjs+1s1;:::;s+lsl;:::;s+nsn(3.65)Sincetheoriginalmaponlyoperatesinrealspace,butthecurrentbasisconsistsofcomplexcon-jugatepairs,arealbasistjfromtherealandimaginarypartofthecurrentcomplexbasissjisintroducedasfollows[6,cf7.58]:t+j=s+j+sj=2(3.66)tj=s+jsj=2i(3.67)TheassociatedtransfermatrixtotherealbasisisAj=1p20BBB@11ii1CCCA=0BBB@t+jjs+jt+jjsjtjjs+jtjjsj1CCCA(3.68)andtheinverserelationaccordingly[6,cf7.59]:sj=t+jitj(3.69)44A1j=1p20BBB@1i1i1CCCA=0BBB@(s+jjt+j)(s+jjtj)(sjjt+j)(sjjtj)1CCCA(3.70)Theamplitudefjconsistsofs+jsjterms,whicharesummarizedtor2j[6,7.67]usingequation3.69:s+jsj=t+j+itjt+jitj=t+j2+tj2=r2j(3.71)ThetransformationfromthemaptotherealbasiscoordinatestjinNormalFormisconductedasfollows[6,cf7.68]:Mj;NF=AjMjA1j=0BBB@1=21=21=2i1=2i1CCCAt+jitjajeifjt+12+t12;:::;t+j2+tj2;:::;t+n2+tn2=aj20BBB@t+je+ifj+eifj+tjie+ifjeifjt+jie+ifjeifj+tje+ifj+eifj1CCCA=aj0BBB@cosfjsinfjsinfjcosfj1CCCA0BBB@t+jtj1CCCA(3.72)Equation3.72illustratesthepropertiesoftheNormalFormbest,whichconsistsofcircularcurvesinphasespacewithonlywithamplitudedependedtuneshifts.ThisformwasalreadyreferredtoasNormalForminexamplesinsection1.2.6,wherethesolutionswerealreadycirclesinphasespace.Theconstantradiusofthecurvecanbeshownby:M+j2+Mj2=a2jr2j=const:(3.73)Furthermore,theonlyamplitude-dependenttuneshiftsareconstantalongonecurve,whichmakestheNormalFormrotationallyinvariant:MNF=RMNFR1(3.74)45TheNormalFormoutputoftheCOSYINFINITYcalculationwillbeintheformofaTaylorexpansionMj;NF=aj0BBB@t+jcosmj+N2cr2j+N4cr4j:::tjsinmj+N2sr2+N4sr4j:::t+jsinmj+N2sr2j+N4sr4j:::+tjcosmj+N2cr2J+N4cr4J:::1CCCA(3.75)wherer2=t+j2+tj2.Fromthisformthetunesandtuneshiftscanbecalculatedwithf=arccoscosm+N2cr2+N4cr4:::=arcsinsinm+N2sr2+N4sr4:::(3.76)whichtakesthefollowingformintheCOSYINFINITYrepresentation:arccoscosm+r2fr2=mfr2sinmr2cosmf2r22sin3mr42cos2m+1f3r26sin5mr6:::(3.77)TransformingthemapM3toNormalFormcoordinatesyields:NF3=AM3A1(3.78)=120BBB@11ii1CCCA0BBB@t++ite+im+12S+3;new(2;1)t+2+t2t+iteim+12S3;new(1;2)t+2+t21CCCA(3.79)=120BBB@e+im+eim+12S+3;new(2;1)+S3;new(1;2)r2t+ie+imeim+12S+3;new(2;1)S3;new(1;2)r2t+1CCCA(3.80)12i0BBB@e+imeim+12S+3;new(2;1)S3;new(1;2)r2tie+im+eim+12S+3;new(2;1)+S3;new(1;2)r2t1CCCA(3.81)=0BBB@cosm+12ReS+3;new(2;1)r2t+sinm+12ImS+3;new(2;1)r2tsinm+12ImS+3;new(2;1)r2t++cosm+12ReS+3;new(2;1)r2t1CCCA(3.82)46Comparingequation3.82totheNormalFormrepresentationinequation3.72yields:cosf=cosm+12ReS+3;new(2;1)r2(3.83)sinf=sinm+12ImS+3;new(2;1)r2(3.84)usingthearccosorarcsin,respectivelyinaTaylorexpansionatr=0uptothe3rd-orderyields:f=mReS+3;new(2;1)2sinmr2=m+ImS+3;new(2;1)2cosmr2(3.85)Assumingthatallsecondordertermsoftheoriginalmaparezero(U2=0),thetunesf=mr28sinmg(3U3(0;3)+U+3(1;2))+b(3U+3(3;0)+U3(2;1))2a(U+3(2;1)+U3(1;2))(3.86)canbesimplycalculatedusingequation3.61.Inmostapplicationsitisusefultoknowr2jintermsoftheoriginalcoordinates.Thetrans-formationAtransforms(~q;~p)!~t+(~q;~p);~t(~q;~p)andconsistsofthecompositionofallthesingletransformationsfortheNormalForm,namely:A=AmmaxAmmax1:::A2A1A0(3.87)Forthesymplecticexamplecaseupto3rdorderwithb6=0andU2=0thetransformationisalreadyrepresentsaveryextensiveformulawith(t+;t)=A3A1(q;p)=1p2b0@Iåk++k=3(S2jk+;k)eim(k+k)eims+k+sk1A0BBB@1+iaib1iaib1CCCA0BBB@qp1CCCA(3.88)Luckily,alltheseprocedurescanbeconductedfullyautomaticallyinCOSYINFINITY.TheNor-malFormAlgorithmisaveryintenseanalyticprocessthatisonlypracticallyusableduetotheDAbasedimplementationinCOSYINFINITY.47CHAPTER4PROTRACTINGCALCULATIONSINPERTURBATIONTHEORYToshowhowpainfulananalyticapproachtosolvingaperturbedharmonicoscillatorsystemcanbe,thefollowingsectionwillpresentsuchanexample.Inthecourseofthecalculationthetediousnessandtheincreaseincomplexityshouldbecomeapparent,eventhoughbothexamplesareontherathernon-complexsideofthedifspectrum.Onemainprocessoftheapproachisthemethodcalled'variationofparameters',whichisintroducedinthefollowingsubsection.4.1VariationofparametersThevariationofparametersisawell-knowntechniquetoobtainsolutionsofaninhomogeneousODEgiventhesolutiontothehomogeneouscase.Therefore,thefollowinginhomogeneousdiffer-entialequationwithalinearhomogeneouspartisconsidered:~z(t)=‹L~z(t)+~f(t)(4.1)where‹Listhelinearcouplingmatrixand~fisthenonlinearinhomogeneity.Thesolutiontothelinearhomogeneousproblem~z(t)=‹L~z(t)isassumedtobeknown:~zhom(t)=‹Z(t)~c(4.2)where~cisjustaparameter,thatcanpotentiallyrepresenttheinitialconditionsand‹Z(t)isthetimedependentsolutionmatrix.SubstitutingthissolutionbackinthehomogeneousODEyieldstheconnectionofsolutionmatrix‹ZandtheODE-couplingmatrix‹L:~zhom(t)=‹Z(t)~c!=‹L~zhom(t)=‹L‹Z(t)~c)‹Z(t)=‹L‹Z(t)(4.3)Tosolveequation4.1,thesolutioninequation4.2isbythemethodofvariationoftheparameter.Inthiscase,theconstantparameter~cisgivenavariation,whichmeansitischangedfromaconstanttoatimedependentvariable~v(t),sothat~zpar(t)=‹Z(t)~v(t)(4.4)48where~zpar(t)representsaparticularsolutiontotheinhomogeneouscase.Usingthisansatzforequation4.1yields:~zpar(t)=‹Z(t)~v(t)+‹Z(t)~v(t)4:3=‹L‹Z(t)~v(t)+‹Z(t)~v(t)=‹L~zpar(t)+~f(t)(4.5)Therefore,‹Z(t)~v(t)=~f(t))~v(t)=Zt0‹Z1(t0)~f(t0)dt0+~v(0)(4.6)Solvingtheintegralmeanssolvingfor~zpar,whereasthecomplexityofthisintegralmayvaryfromtrivialtounsolvable,dependingontheintegrand‹Z1(t)~f(t).Thesolutionforequation4.1ishence:~z=~zhom+~zpar=‹Z(~c+~v(t))(4.7)4.2ExampleofananalyticalperturbationtheoryapproachToillustratethemethodofvariationofparameters,twoproblemsshallbediscussedinthefol-lowingsection.Thesolutionswillbeapproximatedorderbyorder.Theproblemisalmosttrivialbutmakestheprocessclearer.Theactualsolutionhasaverysimpleformandmakesthecomparisontotheorderbyorderapproximationofthemethodpossible.Thesecondexampleisonlyaslightvariationofthetone,whichremovesthesymmetryintheproblem.Thiscausesadramaticincreaseintheanalyticcomplexityconcerningthesolvingoftheintegralfromequation4.1.4.2.1SymmetricperturbedharmonicoscillatorexampleFirst,thesymmetricallyperturbedharmonicoscillator,thatisalreadyknownfromtheexampleintheFlowOperatorsection1.2.6.1,isconsidered.Thesolutiontothisproblemisgivenby490BBB@x(t)p(t)1CCCA|{z}~z(t)=0BBB@cos((1+ar2)t)sin((1+ar2)t)sin((1+ar2)t)cos((1+ar2)t)1CCCA|{z}‹Z(t)0BBB@x0p01CCCA|{z}~c0(4.8)wherer2=x2+p2=x20+p20.UsingthesumangularformulaandtheTaylorexpansionatar2=0thesolutioncanberewrittenasfollows:0BBB@x(t)p(t)1CCCA=0BBB@cos(t)cos(ar2t)sin(t)sin(ar2t)sin(t)cos(ar2t)+cos(t)sin(ar2t)sin(t)cos(ar2t)cos(t)sin(ar2t)cos(t)cos(ar2t)sin(t)sin(ar2t)1CCCA0BBB@x0p01CCCA=0BBB@cos(t)sin(t)sin(t)cos(t)1CCCA0BBB@x0cos(ar2t)+p0sin(ar2t)p0cos(ar2t)x0sin(ar2t)1CCCA=‹Z(t)0BBB@x0+p0ar2tx02(ar2t)2p06(ar2t)3+x024(ar2t)4+O(t5)p0x0ar2tp02(ar2t)2+x06(ar2t)3+p024(ar2t)4O(t5)1CCCA=‹Z(t) ~c0+(ar2t)~d0(ar2t)2~c02(ar2t)3~d06+(ar2t)4~c024+O(t5)!(4.9)Thegoalofthefollowingcalculationis,tostepbystepcalculatethetermsoftheequationabovetoshowhowtheperturbationmethodapproachesthesolution.TheapproachstartsoffwiththeequationsofmotionswhicharegivenbytheHamiltoniananditsHamiltonequations:asfollows:Ha=p22+x22+a4p2+x22(4.10)0BBB@qp1CCCA|{z}~z(t)=0BBB@¶Ha¶p¶Ha¶q1CCCA=0BBB@01101CCCA|{z}‹L0BBB@qp1CCCA|{z}~z(t)+a0BBB@p3+x2px3p2x1CCCA(4.11)50First,thesolutiontothehomogeneouspartofthedifferentialequation~z(t)=‹L~z(t):0BBB@xhom(t)phom(t)1CCCA|{z}~zhom(t)=0BBB@cos(t)sin(t)sin(t)cos(t)1CCCA|{z}‹Z(t)0BBB@x0p01CCCA|{z}~c0(4.12)isrequired.Afterwards,thewell-knownansatzx1(t)=xhom(t)+Dx1(t)(4.13)p1(t)=phom(t)+Dp1(t)(4.14)issubstitutedintheinhomogeneousODEfromequation4.11whichyields:~zhom+D~z1=~z1=‹L~z1+a0BBB@p31+x21p1x31p21x11CCCA(4.15)=‹L(~zhom+D~z1)+a0BBB@p3hom+x2homphomx3homp2homxhom1CCCA+aO(Dx1;Dp1)(4.16)Fortheordercalculation,onlytermslineartoDx1;Dp1oraareconsidered,therefore:0BBB@Dx1(t)Dp1(t)1CCCA|{z}D~z1(t)=0BBB@01101CCCA|{z}‹L0BBB@Dx1(t)Dp1(t)1CCCA|{z}D~z1(t)+a0BBB@p3hom+x2homphomx3homp2homxhom1CCCA|{z}~f0(t)(4.17)Tomaketheintegrationfromthevariation-of-the-parametermethodaseasyaspossibleitisuseful51tosimplify~f0:~f0(t)=0BBB@(p0costx0sint)3+(x0cost+p0sint)2(p0costx0sint)(x0cost+p0sint)3(p0costx0sint)2(x0cost+p0sint)1CCCA(4.18)=a0BBB@p0costx0sintx0costp0sint1CCCA(x0cost+p0sint)2+(p0costx0sint)2(4.19)=a‹Z(t)0BBB@p0x01CCCAp20+x20=ar2‹Z(t)~d0(4.20)Fromequation4.6itisknown,thatD~z1canbecalculatedwiththevariationoftheparameterasfollows:D~z1(t)=‹Z~v(t)=‹ZZt0‹Z1~f0(t0)dt0=‹Zar2~d0Zt0‹Z1‹Z(t)dt0=‹Zar2~d0t(4.21)Therefore,thecombinedsolutionfor~z1is:~z1=~z0+D~z1=‹Z~c0+ar2~d0t(4.22)whichisthepartofthesolutioninequation4.9upto1st-orderina.Forthe2nd-orderperturbationanequivalentansatztotheoneisused:x2(t)=x1(t)+Dx2(t)(4.23)p2(t)=p1(t)+Dp2(t)(4.24)BysubstitutingthisansatzintheHamiltonequationsthefollowingdifferentialequationsareob-tained:52~zhom+D~z1+D~z2=~z2=‹L~z2+a0BBB@p32+x22p2x32p22x21CCCA(4.25)D~z1+D~z2=‹LD~z1+~f1+D~z2(4.26)=‹L(D~z1+D~z2)+a0BBB@p31+x21p1x31p21x11CCCA+aO(Dx1;Dp1)(4.27)whichcanbesummarizedto0BBB@Dx2(t)Dp2(t)1CCCA|{z}D~z2(t)=0BBB@01101CCCA|{z}‹L0BBB@Dx2(t)Dp2(t)1CCCA|{z}D~z2(t)+a0BBB@p31+x21p1x31p21x11CCCA|{z}~f1(t)a0BBB@p3hom+x2homphomx3homp2homxhom1CCCA|{z}~f0(t)(4.28)Fromthis,thegeneralpatternbecomesapparent:~zn+D~zn+1=‹L(~zn+D~zn+1)+~fn+1(4.29)D~zn+1=‹LD~zn+1+aO(Dxn+1;Dpn+1)+nåi=0(1)n+i~fi(t)(4.30)with~fi=a0BBB@p3i+x2ipix3ip2ixi1CCCAi=0=a0BBB@p3hom+x2homphomx3homp2homxhom1CCCA(4.31)Therefore:D~zn(t)=‹LD~zn(t)+(1)nnåi=1(1)i~fi(t)(4.32)OnceagainalltermsinvolvinganatogetherwithaDxn+1orDpn+1areneglectedfortheperturbationorderandmarkedasO(Dxn+1;Dpn+1).Tothesecondorderapproximation,theinhomogeneity~f1fromequation4.28canbebyusingthesolutionfor~f0fromequation4.20andreplacingthecomponents(x0;p0)in~c0bythecomponents(x00;p00)of~c00=~c0+ar2~d0tfromequation4.22:53~f1(t)=a0BBB@p31+x21p1x31p21x11CCCA=a‹Z0BBB@p00x001CCCAp020+x020(4.33)=a‹Zr2(a2r4t2+1)0BBB@p0ar2tx0(x0+ar2tp0)1CCCA(4.34)=a‹Zr2(a2r4t2+1)~d0ar2t~c0(4.35)thetotalinhomogeneouspartistherefore:~f1~f0=‹Za3r6t2~d0ar2t~c0a2r4t~c0(4.36)Integratingaccordingtoequation4.6yields:~v2(t)=(ar2t)2~c02+(ar2t)3~d03(ar2t)4~c04(4.37)~z2(t)=‹Z(t)(~c0+~v1(t)+~v2(t))(4.38)=‹Z(t) ~c0+(ar2t)~d0(ar2t)2~c02+(ar2t)3~d03(ar2t)4~c04!(4.39)whichagreeswiththeexactsolutioninequation4.9nowupto4th-orderina.Forthen2-orderina,theprocessisequivalent.Beginningwiththeansatzxn(t)=xn1(t)+Dxn(t)(4.40)pn(t)=pn1(t)+Dpn(t)(4.41)andsolvingtheinhomogeneousODED~zn(t)=‹LD~zn(t)+(1)nnåi=1(1)i~fi(t)(4.42)bysolvingtheintegralgiveninequation4.6.544.2.2AsymmetricperturbationInthisnextexample,theprocessbecomesalotmoretedious,duetothesymmetrybreakintheHamiltonian.ThenewsystemisgivenbythefollowingHamiltoniananditsHamiltonequations:H=p22+x22+ax44(4.43)with0BBB@xp1CCCA=0BBB@¶Ha¶p¶Ha¶x1CCCA=0BBB@01101CCCA|{z}‹L0BBB@xp1CCCA|{z}~z(t)0BBB@0ax31CCCA(4.44)usingthesameansatzx1(t)=xhom(t)+Dx1(t)p1(t)=phom(t)+Dp1(t)asabove,theHamiltonequationsinorderperturbationcanbederivedasfollows:0BBB@x1p11CCCA=0BBB@p1x11CCCA0BBB@0ax311CCCA=0BBB@01101CCCA0BBB@x1p11CCCA0BBB@0ax3hom1CCCA+aO(Dx1)(4.45)0BBB@Dx1Dp11CCCA=0BBB@01101CCCA0BBB@Dx1Dp11CCCA0BBB@0ax3hom1CCCA(4.46)Followingthemethodofvariationoftheparameter,thefollowingintegraloftheinhomogeneityhastobesolved:~v1(t)=aZt0‹Z1(t)~ep(x0cost+p0sint)3dt(4.47)=aZt00BBB@sintcost1CCCA(x0cost+p0sint)3dt(4.48)55Eventhoughthecalculationbyhandispossible,acomputeralgebraprogramasitisimplementedinWolframAlphaR,wasusedtocalculatethefollowingsolution:~v1(t)=a320BBB@p012r2t+(p203x20)sin4t8p20sin2tx0(12r2t+(x203p20)sin4t+8x20sin2t)1CCCA(4.49)a320BBB@4x0(3p20+x20)cos2tx0(x203p20)cos4t4p0(p20+3x20)cos2t+p0(p203x20)cos4t1CCCA(4.50)Theordersolutionisthen~z1=‹Z(~c0+~v1(t)).Thismethodproducesalreadyquiteextensivecalculationsinthestep.Foreveryfollowingstepevenmoreextensiveintegralsoftheform~vn(t)=aZt00BBB@sintcost1CCCAn1åi=0(1)n+ix3idtwherexi=xhomfori=0,havetobesolved.Consideringthattheproblemsarestillratherbasic,butthecalculationsarealreadyveryextensive,theneedforadifferentapproachbecomesapparent.Thefollowingchapter,therefore,introducesacomputer-basedapproachthatisgenerallyapplica-blefortimeindependentperturbationtotheharmonicoscillator.TheDAframework,whichisimplementedintheCOSYINFINITYprogramusedforthecalculationmakesanautomaticcalcu-lationofthesolutionuptoarbitraryorderpossible.56CHAPTER5PERTURBEDHARMONICOSCILLATORTheHarmonicOscillatorisoneofthebasicmodelsthatisusedinmanyofphysicsandthereforeisofgreatimportance.Inmostcases,theperiodicmodelconsidersaHarmonicoscillatorwithparameter-dependentperturbations.AperturbedharmonicoscillatorcanberepresentedbythesumoftheHamiltonianoftheunperturbedcaseH0andtheperturbationterm:H=H0+Hper.Theperturbationtermcangenerallyconsistofmultipleparameter-dependentterms.5.1ThePendulumInthissection,thePendulumoscillationisconsideredasaperturbationtothesmallangleapprox-imation,whichrepresentstheunperturbedclassicharmonicoscillator.TheexampleshallillustratehowtheDAframeworkinCOSYINFINITYtogetherwithimplementedtheNormalFormAlgo-rithmcanbeusedtoapproximatetheamplitudeandpossiblyparameter-dependenttuneshiftsofaperturbedHamiltonian,numericallyasanalgebraicexpressiontheinformofapolynomial.5.1.1IntroductiontotheProblemConsideringamathematicalPendulumofwithpointmassmatalengthlfromthepivotpointinaconstantgravitationalwithgravitationconstantg.ThePendulumenclosesanangleqwiththeverticalaxisasillustratedin5.1.Therefore,theLagrangianinthegivencoordinatesis:L=ml22q2ml2w20(1cos(q))(5.1)wherew20=gl.TheHamiltoniancanbederivedasfollowsinthegeneralizedcanonicalcoordinateandmomentum(q;p):H=qpqL=q¶L¶qL=ml22q2+ml2w20(1cos(q))(5.2)H=p2q2ml2+ml2w20(1cos(q))(5.3)57Figure5.1IllustrationshowsmathematicalPendulumoflengthlwithpointmassminagravitationalofstrengthg[10].UsingtheHamiltonequations,theorderODEcanbederived:0BBB@qp1CCCA=0BBB@¶H¶p¶H¶q1CCCA=0BBB@01ml2ml2w2001CCCA0BBB@sin(q)p1CCCA(5.4)=0BBB@01ml2ml2w2001CCCA0BBB@qp1CCCAml2w200BBB@0q33!+q55!q77!+:::1CCCA(5.5)TheDANormalFormAlgorithmrequiresanon-zeroconstantlineartermofthemaptobeabletogeneratecertaininversefunctionswithintheprocess.Asalreadymentionedinsection1.2.3,theDAframeworkonlyworkswithinaring,sincetheinverseelementisonlyforelementswithanon-zeroconstantpart.Hence,atransformationtocoordinatesinwhichthesystemhasalinearpartthatisnotpurelyparameter-dependent.Fortheinvestigationofthistransformationitis58helpfultoconsidertheunperturbedharmonicoscillator,wherenononlineartermsappearandthesolutionisknown.5.1.2UnperturbedcaseTosolvetheproblemorapproximateasolution,itisoftenhelpfultoconsidertheunperturbedcaseForthePendulum,thiscaseisknownasthesmallangleapproximation,whereeitherthe1cosqˇq2=2expressionintheHamiltonianorsinqˇqintheHamiltonequationsaresubstituted.WithrespecttofurtherreferencesthefollowinggeneralunperturbedclassicharmonicoscillatorwithitsHamiltoniananditsHamiltonequationsareH=p22m+mw20q22(5.6)0BBB@qp1CCCA=0BBB@¶H¶p¶H¶q1CCCA=0BBB@01mmw2001CCCA0BBB@qp1CCCA(5.7)Asuitabletransformationofthe(q;p)-coordinatestomakethelinearpartoftheODEparameterindependentconsistsoftherealandimaginarypartofthetwocomplexconjugateeigenvectorsofthecouplingmatrix:~v=0BBB@imw011CCCA(5.8)Thetransformation‹Tanditsinversearescaledasalreadydiscussedinthediagonalizationsection1.2.4,sothatthedeterminateofthetransformationisofmagnitude1.‹T1=pmw00BBB@01mw0101CCCA‹T=1pmw00BBB@01mw001CCCA(5.9)59ThetransformationisappliedtotheODEinequation5.7:‹T0BBB@qp1CCCA=‹T0BBB@01mmw2001CCCA‹T1‹T0BBB@qp1CCCA(5.10)0BBB@ppmw0pmw0q1CCCA=0BBB@0w0w001CCCA0BBB@ppmw0pmw0q1CCCA(5.11)0BBB@ppmw0pmw0q1CCCA=0BBB@0w0w001CCCA0BBB@ppmw0pmw0q1CCCA(5.12)0BBB@p1q11CCCA=w00BBB@01101CCCA0BBB@p1q11CCCA(5.13)wherethetransformedvariablesare(q1;p1)=pmw0q;ppmw0.Thistransformationpre-servesthesymplecticstructureoftheHamiltoniansincethenewcoordinatesalsosatisfythePois-sonbracketcondition:fq1;p1g=1(5.14)Hence,thetransformationisacanonicaltransformationwhichpreservingtheformoftheHamiltonequations.ThisisespeciallyessentialforthetransformationoftheperturbedHamiltonianslateron.Inadditiontothecoordinatetransformation,ascalingofthetimettot0=w0tisnecessarytoaccomplishaparameterindependentlinearpart.Therefore,dt0=1w0dtandthetransformedHamiltonequationsyield:0BBB@p1q11CCCA=0BBB@01101CCCA0BBB@p1q11CCCA(5.15)60ThisformoftheODEissuitablefortheDAbasedNormalFormAlgorithmimplementedinCOSYINFINITY.ThesolutiontothisODEisalreadyknownfromvarioustextbooks:0BBB@p1q11CCCA=0BBB@cos(t0)sin(t0)sin(t0)cos(t0)1CCCA0BBB@¯p1¯q11CCCA(5.16)where(¯q1;¯p1)=pmw0¯q;¯ppmw0istheinitialstateatt0=0.Theexampleinsection1.2.6alsosolvestheODE1.64,whichisequivalenttoequation5.15for1+ar2=1usingtheFlowOperator.Notethatequation5.16isalreadyinNormalFormcoordinatesandthereforedoesnotrequireanyfurtherDAmanipulation.Aftertransformingbacktothe(q;p)-systemandrescalingthetimet0=w0t,theresultisgiveninitswell-knownform:‹T10BBB@p1q11CCCA=‹T10BBB@cos(t0)sin(t0)sin(t0)cos(t0)1CCCA‹T‹T10BBB@¯p1¯q11CCCA(5.17)0BBB@qp1CCCA=0BBB@cos(w0t)sin(w0t)mw0mw0sin(w0t)cos(w0t)1CCCA0BBB@¯q¯p1CCCA(5.18)Thetune(unperturbedfrequencyofoscillation)inthisexampleissimplyw0.NotethatE0=p22m+mw20q22=¯p22m+mw20¯q22=const:(5.19)isalsofor¯p=rsin(j0),¯q=rmw0cos(j0)withr=p2E0m.Thereforethesolutionin61equation5.18canberewrittento:0BBB@qp1CCCA=0BBB@¯qcos(w0t)+¯psin(w0t)mw0¯qmw0sin(w0t)+¯pcos(w0t)1CCCA(5.20)=0BBB@rmw0cos(w0t)cos(j)rsin(w0t)mw0sin(j)mwrmw0sin(w0t)cos(j)rcos(w0t)sin(j)1CCCA(5.21)=0BBB@rmw0cos(w0t+j)rsin(w0t+j)1CCCA(5.22)Thisderivationwillbehelpfullateroninsection5.3.Afterthisextensiverevisitoftheunperturbedclassicharmonicoscillator,acollectionofvariousformsoftheunperturbedsolutionsandthecanonicaltransformationneededtomakethelinearpartofthePendulumequation5.4parameter-independenthavebeenderived.5.1.3PendulumtransformationFromtheunperturbedcaseabove,thetransformation‹Tanditsinverseareknownandgiveninequation5.9.ThesametransformationcanbeappliedtothePendulumequation5.4withm0=ml2,whichyieldsthefollowingdifferentialequation:0BBBB@pqml2w0pml2w0q1CCCCA=0BBBB@ml2w20qml2w0sin(q)pml2w0pml21CCCCA=0BBBB@ml2w20qml2w0sin q1qml2w0!w0p11CCCCA(5.23)0BBB@p1q11CCCA=w00BBBB@q1pml2w0 sin q1qml2w0!q1qml2w0!p11CCCCA(5.24)62withthetransformation(q;p)= q1qml2w0;pml2w0p1!andtheappropriatescalingofthetimetot0=w0t,thefollowingformwithlinearparameter-independenceisgiven:0BBB@p1q11CCCA=0BBB@01101CCCA0BBB@p1q11CCCA+0BBBB@q313!ml2w0q515!ml2w02+q717!ml2w03:::01CCCCA(5.25)=0BBB@01101CCCA0BBB@p1q11CCCA+0BBB@aq313!a2q515!+a3q717!a4q919!+:::01CCCA(5.26)fortheimplantationinCOSYINFINITYtheparametera=1=ml2w0isintroduced.ThestepoftheinCOSYINFINITYimplementedprocedureisthecalculationofatransfermapMfromt0=0tot0=1regardingthePendulumODE5.26,withoneoftheDAbasedintegratorsmentionedabove.AnillustrationofthetransfermapusingRK4withh=0:001isin5.2.Figure5.2TheshowsthephasespacecurvesofthePendulumoscillationaccordingtothetransfermapatt0=1,whichwasgeneratedbyintegratingequation5.4withtheDAbasedRK4in1000stepsofstep-sizeh=0:001.Fortheillustration,allparametersweresetto1.Thetransfermapwastrackedfor1000iterations.Thedifferentcurvesrepresentthefollowinginitialconditions,listedfrominnertooutercurve:q=:3;:6;:9;1:2;:::;3:0;p=0.Detailsregardingtheseeminglyclosedandfragmentedcurvescanbefoundinsection5.2.63Inthenextstepoftheprocedure,thetransfermapistransformedbyAintotheNormalFormcoordinates(t+;t)usingtheDANormalFormAlgorithmintroducedinthechapterabove.MNF.MNF=AMA1(5.27)ThetransfermapinNormalFormcoordinatesisillustratedinTheNormalFormcoor-Figure5.3TheshowsthephasespacecurvesofthePendulumoscillationaccordingtothetransfermapatt=1,whichwasgeneratedbyintegratingequation5.4withtheDAbasedRK4in1000stepsofstep-sizeh=0:001withordertruncation20.Fortheillustration,allparametersaresetto1.Thetransfermapatt=1wastransformedtoNormalformcoordinatesandtrackedfor1000iterations.Thedifferentcurvesrepresentthefollowinginitialconditions,listedfrominnertooutercurve:q=:3;:6;:9;1:2;:::;3:0;p=0.Thecurvesshowcircularmotionandtheseparationinthefragmentedcurveshasaconstantdistanceincontrastto5.2.Detailsregardingthisareinsection5.2.dinatesrevealthedesiredtuneshiftsusingequation3.76.Scaledtotheoriginaltimet=w0t0thetuneshiftsaregivenasfollows:w=w0 1ar2163a2r410245a3r616384165a4r8222189a5r10225Oa6r12!(5.28)64NotethattheactualCOSYINFINITYcoefaregiveninthescaledtimeoft0,whichmeanswithoutthew0.Additionally,thecoefaregiveninpointnumberrepresentationinbase10asshownintable5.1.Thefractionsinequation5.28wereonlyusedforillustrativepurposes,butagreewiththeCOSYresultwithanerror<1014,whichisthemagnitudeofpointaccuracy.Table5.1CoefoftheCOSYresultuptoorder10inrforthePendulumtuneswt0(r)showninequation5.28withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.FactorCoefFractionDifferenceinrepresentation10.99999999999999081<9:21015ar2-.6249999999999629E-01116=24<3:81015a2r4-.2929687500000584E-0231024=3210<5:91016a3r6-.3051757812506449E-03516384=5214<6:51016a4r8-.3933906555075291E-041654194304=165222<1:11015a5r10-.5632638932827286E-0518933554432=189225<1:61015Forapplicationsthequantitywintermsofr2isnotveryusefulaspresentedinequation5.28.ThegivenNormalFormtransformation:A:(q1;p1)!t+;tfromequation5.27andequation3.71,makeitpossibletorepresentr2intermsoftheknownquantities(q1;p1).ThecorrespondingCOSYINFINITYtermsandcoefforr2(q1;p1)canbefoundinAPPENDIXtableA.1aswellaswt0(q1;p1),whichisintableA.2.Ingeneral,wt0(q1;p1)istheCOSYINFINITYapproximationtotheproblem,butinthisspecialcasethetransformation(q1;p1)backintothe(q;p)-systemcanbeeasilypreformedbyCOSYINFINITY.Thediflieswithintherelation(q1;p1)=pmw0q;ppmw0=qpa;pap,whereneither1=xnorpxarepossibleoperationsintheDAarithmetic.Hence,asecondvariableb=1=awiththepropertythatab=1isintroduced.Thesquare-rootissueisinthiscasenotrelevant,sinceq1andp1onlyoccurwithevenexponentsandtherefore:q21=bq2p21=ap2withb=1a=ml2w0(5.29)65Notethattheorderisraisedduetothesubstitutioninequation5.29.Foreachtwoordersofq1orp1respectively,cometwoordersofqorpandoneorderofaorb,respectively,whichwillautomaticallyresultinorderlossforhigherordersduetotheordertruncationoftheprocess.Thefollowingsubstitutionsolvesthisproblem,bysimplyintroducingq0=q2andp0=p2tokeeptheorderoftherespectivetermsconstant.Possibleadditionalprocedures,thathavenotyetbeenimplemented,couldcancelalltermswithambn!ammin(m;n)bnmin(m;n)(5.30)resultinginatleastoneoftheexponentsofaorbtobezeroforeachterm.AtablewiththeCOSYINFINITYoutput,forwt0(q;p)inthet0timeframe,canbefoundintableA.3intheappendixandthetermsinthefractionalapproximationarewt0(q;p)= 1q2+a2p216+q430725p410245a2p2512+:::!(5.31)TheperiodTcanbecalculatedbytakingtheinverseoftheequationaboveaccordingtoequation3.30.TheInversionispossibleintheDAframework,duetothenonzeroconstantpartofwt0=1+f(q;p).Hence,TisT=2pw=2pw01wt0=2pw011+f(q;p)(5.32)TheresultforTdisregardingtheprefactorof2pwisgivenintable5.2.Thetermsinthefractionrepresentationarewritteninequation5.33.T=2pw0 1+q2+a2p224+11q43210+9q2a2p229+9a4p4210+:::!(5.33)Sincethepair(q;p)representsanypointonthephasespacecurveofthemotion,asetofinitialconditions(q0;p0)=(q0;0)canbechosentosimplifytheequationforT.Theamplitudeq0oftheoscillationcanbederivedfromanysetofinitialconditionsasfollows:q0=arccos cos(q0)p202m2l4w20!(5.34)For(q;p)=(q0;0),Tyieldsthefollowingresult:T=102pw0 1+q2016+11q403072+173q60737280+22931q801321205760+1319183q100951268147200!(5.35)66Table5.2CoefoftheCOSYresultforthePendulumPeriodTt0(q;p)showninequation5.33withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.FactorCoefFractionDifferenceinrepresentation10.99999999999999961<4:01016q20.6250000000000076E-01124<7:61016a2p20.6250000000000079E-01124<7:91016q40.3580729166662193E-02112103<4:51015q2a2p20.1757812499999105E-013329<9:01015a4p40.8789062499995597E-0233210<4:51015q60.2346462673688346E-03173214325<7:81015q4a2p20.3112792968767812E-02317214<1:81014q2a4p40.4577636718776126E-02352214<2:71014a6p60.1525878906257389E-0252214<7:41015q80.1735611567867274E-04239972223257<4:51015q6a2p20.4541397095222981E-0323812205<5:01014q4a4p40.1370906829873207E-025323221<4:01014q2a6p60.1168251037599050E-025272220<1:41015a8p80.2920627594054748E-035272222<6:11015q100.1386762528983589E-051773106322634527<2:31014q8a2p20.5999931270428728E-0453265922657<3:11014q6a4p40.3120799860718536E-035611032253<1:41013q4a6p60.4940728349303492E-03573292253<2:81013q2a8p80.2957135440714151E-0334725226<1:81013a10p100.5914270880263635E-043472226<2:31014Notethattheexactsamecoeffromtable5.2wereusedfor(q;p)=(q0;0).ThisresultcoincideswiththegeneralanalyticformulaforthePendulum:DTT0=¥ån=1 (2n)!22n(n!)2!2sin2n q202!=q2016+11q403072+173q60737280+:::(5.36)whichisgiveninequation(8)of[20],withT0=2pw0andDT=TT0.TheCOSYresultswere67thereforeaccurateuptothetinypointerrors<2:31014.Theapproximationtofractionseemsreasonableforthosesmallerrors.ConsideringgrandpasPendulumclock,whichoscillateswithacertainperiodT1andamplitudeq0,howdoestheperiodchange,withsmallvariationstotheamplitude?Therelativeerrorintheperiodisasfollows:DTT1=T(q0+Dq)T(q0)T(q0)(5.37)Figure5.4illustratesthedependenceoftherelativeerrorintheperiodonq0andDq.AssumingthatthePendulumissupposedtooscillatewithanamplitudeofq0=p6=30degandisoffsetbysq=p36=5deg,therelativeerrorisapproximately0:63%,whichisalready9min=day.EvenworseistocalculatetheperiodT0=2pqlgofthePendulumfromthesmallangleapproximation,comparedtotheactualperiodatanamplitudeofq0=p6=30deg,whichyieldsarelativeerrorof1:74%,whichisapproximately25min=day.00:020:040:060:080:10:120p12p6p4p35p12p2DT=T1q0Dq=1p=36Dq=2p=36Dq=3p=36Dq=4p=36Dq=5p=36Dq=6p=36Figure5.4Thegraphillustratestheamplitudeq0dependenceoftherelativeperiod-errorfordifferentamplitudeshiftsDq.685.2NormalFormUniquenessThefollowingsectionlargelydrawsfrom[2].LookingattheillustrationsofthetransfermapinthePendulumcalculation5.2and5.3),itbecomesapparentthatbothshow'fragmented'andseeminglycontinuouscurves.Thereasonforthisbaressomeuniqueproperties.Witheveryitera-tionofthetransfermapMt0,thestateismappedonetimestepoft0further.Sinceitisaperiodicsystem,thenumberofiterationsittakesfortheiterationtoreachtheoriginalstartingpointcanbecalculated.IftheperiodofthecurveT(r)dividedbythetimestep-sizeofthetransfermapt0canbeexpressedasafractionofintegervalues(arationalnumber),thenthenumeratordeterminestherequirednumberofiterationsandthedenominatordeterminesthenumberofrevolutionsrequiredbeforeendingupatthestartingpointagain.T(r)t0=IterationsRevolutions=nk(5.38)Inthiscase,theiterationresonates,whichmeansthatonlycertainpointsonthecurvearereachedandformaclosedsystem.Thesepointsareperiodnedpoints,whichcanberelatedtotheassociatedamplitudedependentfrequency,whichisfortheparticularcurve.IftheratiooftheamplitudedependentPeriodandstep-sizeisnotarationalnumber,theiterationswillneverresonatewithearlieriterationsandthereforenotcreateanyedpoints,whichgeneratestheseem-inglycontinuouscurves.Theperiodnedpointsdonotchangeunderbijectivetransformations.Also,theyformaedpointstructure,whichisadensesubsetofthecompletenessofallcurvesjustliketherationalnumbersformadensesubsetoftherealnumbers.Theinvarianceoftheedpointsconservestheassociatedresonancesandmakesthemaninvariantaswell.TheNormalFormcoordinatesarespecialcoordinates,inwhichthefrequencyalongthephasespacecurveisconstantandthereforerotationallyinvariant.Thismakesthesecoordinatesuniquely'natural'fortheresonanceextraction.Figure5.6illustratesthisbypresentingsome'low'periodedpoints,whichhavethesamedistancebetweeneachedpointofthecurve.Theoriginalrepresentationin5.5doesnothavethatproperty,whichcanbeseenespeciallyfurtherawayfromtheorigin.ThisspecialpropertymakestheNormalFormunique.69Figure5.5Thephasespacecurvesoriginatefromthesamemapusedin5.2.Thedifferentcurvesconsistof'low'periodedpoints,whichrepresenttheresonancesofthecurve.Thedistancebetweenthesingleresonancepointsillustrateshowthefrequencychangesalongthecurve.Figure5.6Thephasespacecurvesoriginatefromthesamemapusedin5.3.Thedifferentcurvesconsistof'low'periodedpoints,whichrepresenttheresonancesofthecurve.IntheNormalForm,thedistancebetweenthesingleresonancepointsisconstantalongonecurve,whichmeansthatthefrequencydoesnotchangealongthecurve.TheNormalFormcoordinatesareuniqueinthisproperty.705.3ComparisontoLieTransformperturbationTheoryJohnCarydiscussedLieTransformperturbationTheoryforHamiltoniansystems[9]onanexam-pleofananharmonicoscillator.TheHamiltonianofthisexamplesystemisgivenin[9,4.18]asfollows:H(p;q)=p22+w20q22+ew20q44+e2aw30q68(5.39)Inthissection,theresultsofthepaper[9]andtheDANormalFormAlgorithmapproachshallbecompared.TheCOSYINFINITYsolutionofwisgivenintermsof(q;p),whilethepaperusesAction-AnglecoordinatesoftheperturbedHamiltonian(J;f).Tomaketheresultscomparable,thestepsinthepaper[9]shallbefollowedtotheminimumextentrequiredtounderstandthetransformationfromtheAction-Anglevariablesbackto(q;p):Inthepaper,theHamiltonianisrewrittenintermsoftheAction-Anglevariablesoftheunperturbedcase(e=0).TheunperturbedHamiltonianshallbecalledH0andisequaltoE0theenergy,asaconstantofthesystem.Fromequation5.22thesolutionoftheunperturbedharmonicoscillatorwithm=1isknowntobethefollowing:p=rsin(w0t+j0)q=rw0cos(w0t+j0)(5.40)r=qp2+w20q2=p2E0(5.41)ThesectiononAction-Anglevariablesandequation1.53showhowtheactionjofthe(e=0)-casecanbederivedasfollows:j=12pIp(E0;q)dq=12pIq2E0w20q2dq(5.42)=p2E02pIs1w20q22E0dq=p2E02pIq1sin2Qdq(5.43)wheresinQ=w0qp2E0andcosQdQdq=w0p2E0.Therefore,dq=p2E0w0cosQanditfollowsforj:j=p2E02pIcosQp2E0cosQw0dQ(5.44)=2E02pw0Icos2QdQ=E0w0(5.45)71comparingtheresultforjintermsofE0withtherelationinequation5.41yields:j=E0w0=r22w0(5.46)withthisresulttheAction-Anglejcanbederivedfromequation1.52:H0=E0=w0j¶H0¶j=w0=j(5.47)j(t)=w0t+j0(5.48)Therefore,theoriginalHamiltoniancanberewrittenintermsoftheunperturbedAction-Anglevariableswithp=p2jw0sinjq=q2jw0cosjisgivenas[9,4.19]h(j;j)=w0j+ej2cos4j+e2aj3cos6j:(5.49)Thefollowingderivationleadingtoequation5.53isshownusing[9,4.30-31+35-37].Theunper-turbedAction-AnglevariablesarethentransformedintothenewAction-Anglevariablestoorderoftheperturbedsystem.Therelationshipbetweenold(j;j)andnewvariables(J;f)isgivenasfollows:0BBB@jj1CCCA=0BBB@fJ1CCCA+eJw00BBB@116sin(4f)+12sin(2f)J8cos(4f)J2cos(2f)1CCCA(5.50)Thesolutionforwupto2ndorderinJisthengivenwithw(J)=2w0+34eJ+e215a165164w0J2(5.51)UsingtheenergyofthesysteminthetransformedvariablesK=3w0J+38eJ2+e2J35a161764w0(5.52)thesolutionforwisgivenupto2ndorderinE:w(E)=2w0+34eEw0+e2Ew0215a165164w0(5.53)72Theresultfromequation5.53iswrittenintermsof(q;p)upto4thordertomakeitcomparabletotheresultfromtheCOSYbasedmethodlateron:w=4w00@1+3e8 q2+p2w20!+e2 q2+p2w20!215aw064212561Aw03e28w20 w20p42+p2q2!(5.54)Thegoalofthissectionistoshow,thattheresultfromequation5.54canbereproducedwithamuchsimplerapproachusingCOSY,whichwouldevenincludethecorrecttermsforp4andfurthermore,allowacalculationoftermsuptoarbitraryorder.FromtheHamiltonianinequation5.39theHamiltonequationsaregivenasfollows:~f=0BBB@qp1CCCA=0BBB@pw20qew20q334e2aw30q51CCCA(5.55)FromthesectionaboveitisknownthatimplementingthisformulainCOSYwouldresultinanerrorintheDANormalFormAlgorithm,sincethelinearpartisonlyparameter-dependent.There-fore,theODEhastobetransformed.FromthePendulumsection(5.1)itisknownthattheformoftheHamiltoniananditsHamiltonequationsisonlyconservedforcanonicaltransformationslikeinequation5.9.Thus,theODEcanbetransformedto:~f=0BBB@p1q11CCCA=0BBB@w0q1eq3134e2aq51w0p11CCCA(5.56)withtheknowntimescalingoft0=w0ttheODEcanbewrittenas:0BBB@p1q11CCCA=0BBB@q1ew0q3134e2aw0q51p11CCCA(5.57)=0BBB@q1a1q3134a2q51p11CCCA(5.58)73wherea1=ew0anda2=e2aw0resultinginthefollowingtunesinthefractionrepresentation:wt0=41+3a1q21+p218+15a2q21+p21264a21 21q41+138q21p21+69p41256!=41+3e8w0q21+p21+e2w20q21+p21215aw064212563e28w20 q21p21+p412!=41+3e8 q2+p2w20!+e2 q2+p2w20!215aw064212563e28w20 w20p42+p2q2!(5.59)theCOSYcoefarelistedintable5.3withthedifferenceoftheresultstothefractionrepre-sentations.Again,theaccuracyisinthemarginofthepointcalculationerror.Table5.3CoefoftheCOSYresultforthetuneswt0(q1;p1)showninequation5.59withcorrespondingfractionrepresentationandmaximalerrorintherepresentation.Atablewithcoeftoequation5.59uptoorder10in(q1;p1)arelistedintheappendixintable5.4.FactorCoefFractionDifferenceinrepresentation10.99999999999999081<9:21015a1q210.374999999999977938<2:31014a1p210.374999999999978038<2:21014a2q410.23437499999999111564<91015a2q21p210.46874999999998231532<1:81014a2p410.23437499999999131564<91015a21q41-.8203125000003507E-0121256<3:51014a21q21p21-.539062500000070869128<7:11014a21p41-.269531250000020469256<2:11014TheCOSYresultfromequation5.59agreeswiththetransformedsolutionfromthepaper[9]inequation5.54,upto4thorderin(q;p)andpointaccuracy.WhiletheCaryapproachrequiresagreatinvestmentincalculationwhichbecomesmoreandmorecomplexwithhigheror-ders,COSYofferstheresultwithpointaccuracyuptoarbitraryorderastable5.4illustrateswithminimaltrade-offsincomputationtime.74Table5.4COSYcoefofwt0(q1;p1)uptoorder14forthecalculationinequation5.59.OrderFactorCoefOrderFactorCoef010.99999999999999012q81a41-0.0249824523925763q21a10.37499999999997712q61p21a41-0.7576446533203203p21a10.37499999999997812q41p41a41-2.0637130737306105q41a20.23437499999999112q21p61a41-1.6848907470704405q21p21a20.46874999999998212p81a41-0.4212226867675885p41a20.23437499999999113q101a1a220.0616836547851376q41a21-0.08203125000003513q81p21a1a221.6304397583009606q21p21a21-0.53906250000007013q71p31a1a220.0000000000000066p41a21-0.26953125000002013q61p41a1a225.1889801025400408q61a1a2-0.11132812500005813q51p51a1a220.0000000000000228q41p21a1a2-1.08398437500023013q41p61a1a226.1447906494150808q21p41a1a2-1.31835937500015013q31p71a1a220.0000000000000068p61a1a2-0.43945312500004313q21p81a1a223.2289505004887109q61a310.03955078124999113p101a1a220.6457901000977429q41p21a310.65771484375003214q101a31a2-0.0729789733886619q21p41a310.92724609375008014q91p1a31a20.0000000000000159p61a310.30908203125001414q81p21a31a2-2.68447494506856010q81a22-0.03936767578128114q71p31a31a2-0.00000000000003410q61p21a22-0.50903320312518714q61p41a31a2-10.32433319091910010q41p41a22-0.93933105468773014q51p51a31a2-0.00000000000008110q21p61a22-0.62622070312510614q41p61a31a2-13.83556365966940010p81a22-0.15655517578128114q31p71a31a2-0.00000000000003511q81a21a20.08450317382810414q21p81a31a2-7.73763656616274011q61p21a21a21.82629394531262014q1p91a31a20.00000000000001511q41p41a21a24.25994873046928014p101a31a2-1.547527313232520TheCOSYresultshallbecomparedtotheearlierresultfromthePendulumbyinsertinge=1=6anda=2=5w0.ThissubstitutionconvertstheHamiltonianfromequation5.39intothe75PendulumHamiltonianfromthesectionbefore(5.1)upto6thorder:H=p22+w20 q22q424+q6720!=6p22+w20(1cos(q))(5.60)substitutingthegivenvaluesforeandainequation5.59yields:wt0=1q2+a2p216+q430725p410245a2p2512+:::(5.61)Thisequationcoincidesinalltermsoforderm6withthePendulumsolutioninequation5.31,whichsupportstheconsistencywithintheCOSYcalculations.ToinvestigatetheCOSYresultabitfurther,thetransfermapistrackedforthefollowingparameters(w0;e;a)=(1;2;2=9)whichcorrespondtoapotentialVintheHamiltonianoftheformshownin5.7.Thepotentialhasthreestablestationarypointsandtwounstablestationarypointsinbetween.Thetrackingofthetransfermapofthesystemfromt=0tot=0:1withtheFigure5.7ThegraphshowsthepotentialV=q22q42+q69oftheHamiltonianinequation5.39fortheparameters(w0;e;a)=(1;2;2=9).Therearethreestablestationarypointsattheoriginandq=q32+p32.Thetwounstablestationarypointsareatq=q32p32.Thepotentialallowsoscillationineachofthethreevalleysaswellasaglobaloscillationoverlargep.parametersmentionedaboveisshownin5.8.Itshowsphasespacecurvesaroundthetheedpoints(stablestationarypointsofthepotential)aswellasaglobalcurveforlargerp.Theconsideredperturbationinthecalculationabovewaswithrespecttotheorigin.Perturbationtheoryingeneralisonlyabletosolveforthedirectsurroundingoftheconsiderededpoint,whichistheunperturbedharmonicoscillatoraroundtheorigininthiscase.Therefore,neitherthesolutionof76Figure5.8ThetransfermapoftheODEinequation5.57wascalculatedusingtheRK4with100stepsofstep-sizeh=0:001untilt=0:1.Theillustrationshowsthephasespacetrackingof1000iterationsusingtheparameters(w0;e;a)=(1;2;2=9).ThecurvesaroundtheoriginaresimilartotheoneinthePendulumexamplein5.2.Forlargerjqjandsmallpthephasespacecurveoscillatesaroundadifferentedpointatapproximately(1:5;0).Forlargeqandpthephasespacecurveoscillatesaroundallthreeedpoints.Figure5.7makesthisbehaviorapparent.theLiePerturbationapproachnortheCOSYresult,yieldthefrequenciesforanyotheroscillationsthataroundtheoriginedpoint.ThetrackingoftheNormalFormofthetransfermapillustratesthispropertyin5.9.IncontrasttotheLiePerturbationtheory,thetrackingpicturesthesurroundingedpoints.TheCOSYapproachcaneasilybevariedinitsinitialconditionstoconsideraperturbationaroundtheedpointat(q0;p0)=q32+p32;0.Thetrackingpictureoftheshiftedsystemwiththesameparametersisshownin5.10.Itisapparent,thattheedpointat(q0;p0)=q32+p32;0isnotdepicted.Itseemslikethetransfermapcanonlydetermineedpointsinthedirectsurroundingoftheconsiderededpoint.TheNormalFormtothisedpointperturbationrevealsthetuneshiftswithrespecttotheunperturbedcase.Table5.5liststhecoefforthetuneswt0(q0;p).Directlyattheedpoint,thefrequencyoftheoscillationisatleastdoublethefrequencyofthe77Figure5.9Trackingof1000iterationsusingtheparameters(w0;e;a)=(1;2;2=9)ofthetransfermapusedin5.8inNormalForm.Incomparisonto5.8onlythecurvesaroundtheoriginarepreserved,whichillustrateshowperturbationtheoryonlyworksinthedirectsurroundingoftheconsiderededpoint.Figure5.10Trackingpictureofthesametransfermapusedin5.8onlywithashiftedreferencepointfortheperturbation.unperturbedcaseattheorigin,whichcanbedeterminedbytheconstanttermofthefrequencies.Alreadysmallvariationstoq0makethefrequencydropalotfasterthanintheorigin-relatedcase,whichcanbedeterminedbythelargenegativecoeffortermsinq0.Ingeneral,theexampleshowshowadaptableandeffectivetheCOSYapproachis.78Table5.5Thetablerevealsthetermsandrelatedcoefofwt0(q0;p)fortheoscillationaroundtheedpoint(q00;p0)=q32+p32;0oftheHamiltonianinequation5.39withtheparameters(w0;e;a)=(1;2;2=9)andq0=qq0.CoefOrderFactorCoefOrderFactor2.33754178896010E+00011.21382387121546E-126q05p-5.16315324537914E+002q02-5.45104992215302E+016q04p2-9.44922625720420E-012p2-1.45822595527180E-136q03p3-9.47032596684882E+003q03-8.60580684709545E+006q02p43.23158337141804E-123q0p2-1.90397253907876E-146q0p5-2.27378667458620E+014q04-5.24990654350866E-016p6-5.94074588085554E+004q02p2-5.58506174494210E+027q07-5.43615977456575E-014p44.99636191467564E-127q06p-6.14780725235178E+015q05-1.74730289073416E+027q05p23.15815384599474E-135q04p1.69335451054548E-127q04p3-1.08965969639085E+015q03p2-1.57848880666779E+017q03p47.53952744185456E-145q02p3-1.20261466986781E-137q02p53.69557903192260E-125q0p45.32068881682924E-127q0p6-1.81373221158553E+026q065.4SolvingperturbedHarmonicoscillatorwithperharmosc.foxAfterthosetwoexamplesofusingtheDAFrameworkinCOSYINFINITY,thegeneralalgorithminperharmosc.foxforanarbitrarytime-independentperturbedHarmonicoscillatorofthefollowingformshallbeexplained:H=a0q22+akqk+b0p22+blpl+ciqjpn(5.62)wherethetermsofakqk,blplandciqjpncanoccurforvariousk's,l'sandcombinationsof(j;n)aslongask>2,l>2andj+n>2forallterms.Tousethealreadyestablishedframework,thefollowingvariablesarea0=mw20andb0=1m.Rewritingtheequationsyields:m=1b079andw20=a0b0.Hence,mw0=qa0b0.TheHamiltonequationsarethereforegivenasfollows:0BBB@qp1CCCA=0BBB@¶H¶p¶H¶q1CCCA=0BBB@b0p+lblpl1+nciqjpn1a0qkakqk1+jciqj1pn1CCCA(5.63)=0BBB@01mmw2001CCCA0BBB@qp1CCCA+0BBB@lblpl1+nciqjpn1kakqk1+jciqj1pn1CCCA(5.64)Theknowntransformation‹Tfromequation5.9andthetimescalingtot0=w0t,tounparame-terisethelinearpartofthedifferentialequationthenyield:0BBB@ppmw0pmw0q1CCCA=0BBB@a0q+kakqk1+jciqj1pnpmw0pmw0b0p+lblpl1+nciqjpn11CCCA(5.65)0BBB@p1q11CCCA=0BBBB@a0q1mw0kakqk11(mw0)k2j(mw0)nj2ciqj1pnmw0b0p1(mw0)l2lblpl11+n(mw0)nj2ciqjpn11CCCCA(5.66)0BBB@p1q11CCCA=w00BBBBBB@0BBB@01101CCCA0BBB@p1q11CCCA+0BBBBBB@kakqk11w0(mw0)k2jciqj1pnw0(mw0)jn2(mw0)l2lblpl11w0+nciqjpn1w0(mw0)jn21CCCCCCA1CCCCCCA(5.67)0BBB@p1q11CCCA=0BBB@01101CCCA0BBB@p1q11CCCA+0BBB@kdkqk11jgiqj1pnlelpl11+ngiqjpn11CCCA(5.68)TheODEinequation5.68isintegratedintoCOSYandtheresultingtransfermapfromtheinitialstateatt0=0tot0=1istransformedtoNormalFormcoordinates,yieldingthetunesinthefollowingformwt0(q1;p1;dk;el;gi)andtheperiodTt0(q1;p1;dk;el;gi).Withthefollowing80substitutions,theresultcanberewrittenintermsoftheoriginalcoordinates.Onestepistobringtherespectivequantitytotheoriginaltimescaling:T(q1;p1;dk;el;gi)=2pw0Tt0(q1;p1;dk;el;gi)(5.69)w(q1;p1;dk;el;gi)=w0wt0(q1;p1;dk;el;gi)(5.70)Thecoordinates(q1;p1)canbetransformedbacktotheoriginalcoordinatesq;p,with(q1;p1)=pmw0q;ppmw0= 4ra0b0q;4sb0a0p!(5.71)Finally,theparametersusedintheAlgorithmcanbescaledbacktotheoriginal:dk=akw0(mw0)k4el=blw0(mw0)l4gi=ciw0(mw0)nj4(5.72)dk=akpa0b0b0a0k4el=blpa0b0a0b0l4gi=cipa0b0b0a0jn4(5.73)TheCOSYresultwillbegivenin(q1;p1;dk;el;gi)andappearsasitisillustratedintable5.6.Thismethodofsubstitutingtheoriginalparametersandvariablesattheendmakesthealgo-rithmmostefbecauseitoperateswiththeminimalamountofordersandthereforeyieldsmaximumprecision.Therearewaystoimplementproblematicparameters,likepaor1=pa,byintroducingadditionalvariablesc=paandd=1=pa,butthatreducestheprecision.Addition-ally,procedureswouldhavetobeimplementedthatassure,thatparametersarecanceledaccordingtocd=1,c2=aandd2=b.Thoseprocedureswouldonlysimplifytheresultatthecostofalowerprecisionandandincreaseincomputingtime.81Table5.6ThetableillustratesanexampleoutputofCOSYfortheperiodTinthe(q1;p1)-coordinatesupto5thorderin(q1;p1).Thecolumn'I'denotestherow-counter.Thecolumnsunder'EXPONENTS'eachrepresentonevariable.Thetwoarethe(q1;p1)-coordinates.Eachadditionalcolumndenotesaparameterdk;elorgistartingwiththeparameterassociatedwiththeperturbationtermenteredtotheprogram.Thenumberintherespectivecolumndenotestheexponentofthevariable/parameter.Thecolumn'ORDER'sumsupalltheexponentsandpresentsthetotalorderoftheterm.Thesecondcolumn'COEFFICIENT'yieldstheCOSYTaylorexpansioncoefregardingtheassociatedterm.ICOEFFICIENTORDEREXPONENTS1-0.841470981100020.54030231101003-0.50509902430104-0.74594703421105-0.55736289412106-0.17208648403107-0.30059578650018-0.61324672641019-0.782284856320110-0.638374156230111-0.305561186140112-6.52E-026050182CHAPTER6CONCLUSIONThisthesisintroducedaveryefapproachtosolvingperiodic,timeindependent,Hamilto-niansystemswithparameter-dependentperturbationsuptoarbitraryorder.ThefoundationofthemethodprovedtobetheDAframeworkimplementedwithinCOSYINFINITY,whichallowsforthecomputer-basednumericalcalculationofalgebraicstructures.Wellknownmethodslikethefourth-orderRunge-KuttacanberealizedintheDAframeworkinanequivalentwaytotheclassicimplementation.Furthermore,integratorswhicharebasedondifferentiationandintegrationsuchastheFloworedpointIntegrator,respectively,canbeimplementedveryefintheDAframework,whichisduetoitsautomaticdifferentiationandanti-differentiationoperations.Asaresult,theDAbasedintegratorsyieldatransfermap,whichisanalgebraicexpressionthatyieldsthestateintermsoftheinitialstate.Thosemapscanalsobeparameter-dependent,incontrasttoregularnumericalintegratorsthatdonotprovideanyalgebraicexpressions.Furthermore,theDAstructurewasusefulintheimplementedDANormalFormAlgorithm,whichtransformedthetransfermapintoNormalFormcoordinatestocalculatethetunes.ThetunesaretheinvariantquantitiesofthesystemanddonotchangealongthephasespacecurveinNormalFormcoordinates,whichmakesthesecoordinatesuniquely'natural'fortheextractionofthetunes.TheNormalFormAlgorithmusesvarioustransformationswithintheprocess,whichalwaysoccurstogetherwiththeirinverses.Inthiscontext,thepropertyoftheDAstructureofonlyyieldinganinversefortermswhichcontainedanon-zeroconstantpartbecameanissueforparameter-dependentmaps.Theoriginaldifferentialequationhadtobetransformedcanonicallytocoordinatesinwhichthelinearparthadanon-zeroconstantterm.Inthelaststep,thecomposi-tionofthesingletransformationsintheNormalFormAlgorithmwhereusedtoexpressthetunesandtheirtuneshiftsintheoriginalcoordinates,tomakethesolutionsuitableforarbitraryinitialconditionsintheoriginalcoordinates.InthecomparisontotheLieTransformperturbationtheory,theadvantageofeasilyexpressing83thesolutionintheoriginalcoordinatesbecameapparent.Additionally,thetrackingpicturesillus-tratedthelimitationsofperturbationtheoryingeneraltothechosenreferencesystem.EventhoughtheCOSYtrackingcouldrevealtheadjacentedpoints,theNormalFormAlgorithmwasonlyabletotransformthephasespacecurvesaroundtheoriginintocircularmotionswithamplitude-dependenttuneshifts.IncontrasttotheLieTransformansatz,asimplecoordinatetransformationtoanotheredpointallowedtousetheNormalFormmethodequivalentlytodeterminethetuneshiftsofoscillationsaroundthenewreferencepoint.Asaresultoftheoverallinvestigation,theprogramperhamosc.foxwaswrittentogenerallysolveparameter-dependentperturbedharmonicoscillators,withtheabilitytochangethereferencepointoftheperturbation.ForthisthesisallproceduresusedwereeitheralreadyimplementedinCOSYINFINITYorwereprogrammedontheUSERlevelsincecomputingtimewasnorelevantfactor.AnimplementationofcertainproceduresintheunderlyingFORTRANcodeasinternalprocedureswouldmakethosedrasticallymoreef84APPENDIX85APPENDIXCOSYCOEFFICIENTSTableA.1COSYcoefofr2(q1;p1)uptoorder10in(q1;p1)forthePendulumcalculation.OrderFactorCoefOrderFactorCoef2q211.00000000000000E+0011q21p61a31.52587890584416E-032p211.00000000000000E+0011q1p71a3-6.23130019805629E-145q41a-5.20833333333330E-0211p81a33.81469726555902E-045q21p21a6.24999999999983E-0214q101a4-3.11449487795811E-075p41a3.12500000000003E-0214q91p1a42.60559018750832E-138q61a24.99131944441055E-0414q81p21a41.73561143225385E-058q41p21a23.58072916659495E-0314q71p31a4-9.96226834659921E-138q21p41a28.78906250007309E-0314q61p41a42.27069853202729E-048p61a22.92968750000307E-0314q51p51a4-2.30420448719341E-1211q81a3-9.93032304185488E-0614q41p61a44.56968944165443E-0411q71p1a32.15106795223296E-1414q31p71a4-5.42693199542332E-1311q61p21a32.34646267818247E-0414q21p81a42.92062760508722E-0411q51p31a32.67750123288113E-1314q1p91a45.23138307236752E-1311q41p41a31.55639648441389E-0314p101a45.84125518994400E-0511q31p51a31.33353180928330E-1386TableA.2COSYcoefofwt0(q1;p1)uptoorder10in(q1;p1)forthePendulumcalculation.OrderFactorCoefOrderFactorCoef019,99999999999990E-0112q41p41a4-5,03063201898618E-043q21a-6,24999999999962E-0212q21p61a4-4,47273254388907E-043p21a-6,24999999999963E-0212p81a4-1,11818313597672E-046q41a23,25520833332584E-0415q101a5-1,47522023408260E-076q21p21a2-9,76562500000149E-0315q81p21a5-1,56177887847264E-056p41a2-4,88281250000031E-0315q71p31a55,01368288248799E-159q61a3-3,11957465283522E-0515q61p41a5-9,92635885946822E-059q41p21a3-1,20035807291816E-0315q51p51a56,26444222558972E-159q21p41a3-2,01416015625245E-0315q41p61a5-1,68214241683875E-049p61a3-6,71386718750670E-0415q31p71a52,25888609846247E-1512q81a4-1,90659174731445E-0615q21p81a5-1,02743506444934E-0412q61p21a4-1,38706631128647E-0415p101a5-2,05487012877491E-05TableA.3COSYcoefofwt0(q;p)uptoorder10in(q;p)forthePendulumcalculation.OrderFactorCoefOrderFactorCoef019,99999999999990E-0112q6p2a2-1,38706631128647E-043q2-6,24999999999962E-0212q4p4a4-5,03063201898618E-043p2a2-6,24999999999963E-0212q2p6a6-4,47273254388907E-046q43,25520833332584E-0412p8a8-1,11818313597672E-046q2p2a2-9,76562500000149E-0315q10-1,47522023408260E-076p4a4-4,88281250000031E-0315q8p2a2-1,56177887847264E-059q6-3,11957465283522E-0515q6p4a4-9,92635885946822E-059q4p2a2-1,20035807291816E-0315q4p6a6-1,68214241683875E-049q2p4a4-2,01416015625245E-0315q2p8a8-1,02743506444934E-049p6a6-6,71386718750670E-0415p10a10-2,05487012877491E-0512q8-1,90659174731445E-0687BIBLIOGRAPHY88BIBLIOGRAPHY[1]VladimirIArnold.Mathematicalmethodsofclassicalmechanics.Translatedfromthe1974RussianoriginalbyK.VogtmannandA.Weinstein.Correctedreprintofthesecond(1989)edition.Springer-Verlag,2ndeditionedition,1978.[2]MartinBerz.Privatecommuncation.[3]MartinBerz.ThenewmethodofTPSAalgebraforthedescriptionofbeamdynamicstohighorders.TechnicalReportAT-6:ATN-86-16,LosAlamosNationalLaboratory,1986.[4]MartinBerz.Themethodofpowerseriestrackingforthemathematicaldescriptionofbeamdynamics.NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipment,258(3):431Œ436,1987.[5]MartinBerz.Differentialalgebraicdescriptionofbeamdynamicstoveryhighorders.Part.Accel.,24(SSC-152):109Œ124,1988.[6]MartinBerz.ModernMapMethodsinParticleBeamPhysics.ACADEMICPRESSAHarcourtScienceandTechnologyCompany,1999.[7]MartinBerz,KyokoMakino,KhodrShamseddine,GeorgHHoffstätter,andWeishiWan.32.COSYINFINITYanditsapplicationsinnonlineardynamics.1996.[8]J.C.Butcher.CoefforthestudyofRunge-Kuttaintegrationprocesses.JournaloftheAustralianMathematicalSociety,3:185Œ201,51963.[9]JohnRCary.LietransformperturbationtheoryforHamiltoniansystems.PhysicsReports,79(2):129Œ159,1981.[10]Chetvorno.Diagramofsimplegravitypendulum,anidealmodelofapendulum.itcon-sistsofamassivebobsuspendedbyaweightlessrodfromafrictionlesspivot,withoutairfriction.whengivenaninitialimpulse,itoscillatesatconstantamplitude,forever.https://commons.wikimedia.org/wiki/File:Simple_gravity_pendulum.svg,Dec2008.[11]SamuelDanielConteandCarlWDeBoor.Elementarynumericalanalysis:analgorithmicapproach.McGraw-HillHigherEducation,1980.[12]E.DCourantandH.SSnyder.Theoryofthealternating-gradientsynchrotron.AnnalsofPhysics,3(1):1Œ48,1958.[13]EtienneForest,JohnIrwin,andMBerz.Normalformmethodsforcomplicatedperiodicsystems.Part.Accel.,24:91Œ107,1989.[14]DavidGoldberg.Whateverycomputerscientistshouldknowaboutarithmetic.ACMComput.Surv.,23(1):5Œ48,March1991.[15]E.HairerandGerhardWanner.SolvingordinarydifferentialequationsII.Springer-Verlag,1991.89[16]GeorgHHoffstaetterandMartinBerz.Rigorouslowerboundsonthesurvivaltimeinparticleaccelerators.ParticleAccelerators,54(2):193Œ202,1996.[17]KyokoMakinoandMartinBerz.COSYINFINITYversion8.NuclearInstrumentsandMethodsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAssoci-atedEquipment,427(1):338Œ343,1999.[18]KyokoMakinoandMartinBerz.COSYINFINITYversion9.NuclearInstrumentsandMeth-odsinPhysicsResearchSectionA:Accelerators,Spectrometers,DetectorsandAssociatedEquipment,558(1):346Œ350,2006.Proceedingsofthe8thInternationalComputationalAc-celeratorPhysicsConferenceICAP20048thInternationalComputationalAcceleratorPhysicsConference.[19]JohnHMathewsandKurtisDFink.NumericalmethodsusingMATLAB,volume31.PrenticehallUpperSaddleRiver,NJ,1999.[20]RobertANelsonandMGOlsson.Thependulum-richphysicsfromasimplesystem.Am.J.Phys,54(2):112Œ121,1986.[21]InstituteofElectricalandElectronicsEngineers.IEEEStandardforBinaryFloating-pointArithmetic.IEEE,1985.[22]GASardanashvily.HANDBOOKOFINTEGRABLEHAMILTONIANSYSTEMS.URSS,2015.90