StudentSeminar Integrable system .pdf

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1、PreprinttypesetinJHEPstyle-HYPERVERSIONStudentSeminar:ClassicalandQuantumIntegrableSystemsGlebArutyunovaaInstituteforTheoreticalPhysicsandSpinozaInstitute,UtrechtUniversity3508TDUtrecht,TheNetherlandsAbstract:Thestudentswillbeguidedthroughtheworldofclassicalandquan-tumintegrablesystems.Star

2、tingfromthefamousLiouvilletheoremand¯nite-dimensionalintegrablemodels,thebasicaspectsofintegrabilitywillbestudiedin-cludingelementsofthemodernclassicalandquantumsolitontheory,theRiemann-HilbertfactorizationproblemandtheBetheansatz.DeliveredatUtrechtUniversity,20September2006-24January2007Co

3、ntents1.LiouvilleTheorem21.1Dynamicalsystemsofclassicalmechanics21.2Harmonicoscillator51.3TheLiouvilletheorem71.4Action-anglevariables92.ExamplesofintegrablemodelssolvedbyLiouvilletheorem112.1Somegeneralremarks112.2TheKeplertwo-bodyproblem122.2.1Central¯eldsinwhichallboundedorbitsareclosed.

4、152.2.2TheKeplerlaws172.3Rigidbody202.3.1Movingcoordinatesystem202.3.2Rigidbodies212.3.3Euler'stop232.3.4OntheJacobiellipticfunctions272.3.5Mathematicalpendulum292.4Systemswithclosedtrajectories313.Laxpairsandclassicalr-matrix323.1Laxrepresentation323.2Laxrepresentationwithaspectralparamete

5、r343.3TheZakharov-Shabatconstruction364.Two-dimensionalintegrablePDEs414.1Generalremarks424.2Solitonsolutions434.2.1Korteweg-de-Vriescnoidalwaveandsoliton434.2.2Sine-Gordoncnoidalwaveandsoliton454.3Zero-curvaturerepresentation474.4Localintegralsofmotion495.QuantumIntegrableSystems555.1Coord

6、inateBetheAnsatz(CBA)565.2AlgebraicBetheAnsatz685.3NestedBetheAnsatz(tobewritten)796.IntroductiontoLiegroupsandLiealgebras80{1{7.Homeworkexercises957.1Seminar1957.2Seminar2967.3Seminar3977.4Seminar4987.5Seminar51007.6Seminar61027.7Seminar71057.8Seminar81071.LiouvilleTheorem1.1Dynamicalsyste

7、msofclassicalmechanicsTomotivatethebasicnotionsofthetheoryofHamiltoniandynamicalsystemscon-siderasimpleexample.LetapointparticlewithmassmmoveinapotentialU(q),whereq=(q1;:::qn)isavectorofn-dimensionalspace.ThemotionoftheparticleisdescribedbytheNewtonequat