Nonintegrability of the two-body problem in consta

arXiv:math/0601382v3[math.DS]8Oct2006Nonintegrabilityofthetwo-bodyprobleminconstantcurvaturespaces∗AlexeyV.Shchepetilov†AbstractWeconsiderthereducedtwo-bodyproblemwiththeNewtonandtheoscillatorpotentialsonthesphereS2andthehyperbolicplaneH2.Forbothtypesofinteractionweprovethenonexistenceofanadditionalmeromorphicintegralforthecomplexifieddynamicsystems.PACSnumbers:02.30.Ik,02.40.Yy,03.65.FdMathematicalSubjectClassification:70F05,37J30,34M35,70H07.1IntroductionThestudyofmechanicsonconstantcurvaturespacesbeguninthenineteenthcenturyaftertheriseofnoneuclideangeometry[1]–[3].SimilarlytotheEuclideancaseinconstantcurvaturesimplyconnectedspaces(thesphereSnandthehyperbolicspaceHn)therearetwoexceptionalcentralpotentialsVNandVo(belowNewtonandoscillatorpotentials).Theyhavesomeniceproperties,whichcanbegroundsfortheirdefinitions.Boththesepotentialsmakeallboundedtrajectoriesofaone-bodyproblemclosed[4].Moreover,thesetrajectories(boundedandunbounded)areconics[5],whichcanbenaturallydefinedinconstantcurvaturespaces[6]–[8].Theone-bodymotionintheNewtonpotentialsatisfiestotheanaloguesofthethreeKeplerlaws[2],[3],[5].ThispotentialisalsothefundamentalsolutionoftheLaplaceequation.ThecorrespondingforceinthehyperbolicspacewasalreadyproposedbyN.Lobachevski(in1835-38)[9]andJ.Bolyai(between1848and1851)[10]asthevalueF(ρ)whichisinversetotheareaofthesphereinH3ofradiusρwithanattractivebodyinthecenter.Theseresultscanbeconsideredaspredecessorsofgeneralrelativity.Aftertheriseofthistheorytheabove-mentionedpaperswerealmostcompletelyforgotten.Similarmodelsattractedattentionlaterfromthepointofviewofquantummechanicsandthetheoryofintegrabledynamicalsystems.Thisleadstotherediscoveryofresultsdescribedaboveinmanypapers,sometimeswithpartialimprovements,seeforexample[11]and[12].NotehoweverthatalmostforgottenresultsofW.KillingandH.Liebmannweredescribedinthesurvey[13].Correspondingquantummechanicalproblemsinconstantcurvaturespaceswerestud-iedin[14]–[18]andotherpapers.Thetwo-bodyproblemwithacentralinteractioninconstantcurvaturespacesSnandHnconsiderablydiffersfromitsEuclideananalogue.Thevariableseparationforthelatter∗c2006IOPPublishingLtd,J.Phys.A:Math.Gen.V.39(2006),pp.5787-5806†DepartmentofPhysics,MoscowStateUniversity,119992Moscow,Russia,e-mailaddress:[e-mail]12Reducedtwo-bodyproblems2problemistrivial,whilefortheformeronenocentralpotentialsareknownthatadmitavariableseparation.Thetwo-bodyproblemwithacentralinteractioninconstantcurvaturespaceswasconsideredforthefirsttimein[19].InEuclideanspacethisproblemisreducedtoaone-bodyprobleminacentralpotentialafterseparatingthecenterofmassmotion.DuetotheabsenceofGalileitransformationsthesituationfortheconstantcurvaturespacesisdifferent.Thetwo-bodyproblemisinvariantwithrespecttotheisometrygroup,butfornon-Euclideanspacethisgroupisnotwideenoughtoimplytheintegrabilityofthisprobleminanysense.Thenaturalproblemoffindingcentralpotentialscorrespondingtointegrabletwo-bodyproblemsisfarfromitssolutionnow.Thiscanbeexplainedbythefactthatexistingmethodsofthetheoryofintegrableandnonintegrabledynamicalsystemsdonotworkinthepresenceofafunctionalparameter.Asalimitingcaseofatwo-bodyprobleminconstantcurvaturespaces,onecanconsidertherestrictedtwo-bodyproblem:the”heavy”bodymoveswithaconstantvelocityalongageodesic,whilethe”light”onemovesinapotentialofa”heavy”body.ThenonintegrabilityofthisproblemwiththepotentialVNandVoonthesphereS2wasprovedin[20],[21]intheclassofmeromorphicfunctions.Similarresultswithsmallerrestrictions,validalsofortherestrictedtwo-bodyproblemonthehyperbolicplaneH2,wereobtainedin[22].Here,weprovethenonexistenceofanadditionalmeromorphicfirstintegralfortherestrictedtwo-bodyproblemonthespacesS2andH2usingtheMorales-Ramistheory[23].2Reducedtwo-bodyproblemsNotethattheclassicaltwo-bodyproblemonSnandHnreachesitsfullgeneralityatn=3[19].ItsHamiltonianreductiontothesystemwithtwodegreesoffreedomwascarriedoutin[19](n=3)andin[24](n=2)byexplicitcoordinatecalculations.Amoreconceptualapproachtothisreductionwasderivedin[25].Hereweshallusethefollowingdescriptionofthereduceddynamicalsystemsforn=2,combiningapproachesfrom[19]and[24]–[26].2.1Thereducedtwo-bodyproblemonthesphereS2LetS2bethesphereoftheradiusRwiththestandardmetric.Theconfigurationspaceforthetwo-bodyproblemonS2isQ=S2×S2
\diag.LetQop≃S2beasubsetofQ,consistingofpairsofoppositepoints.ThephasespaceT∗QcanberepresentedasT∗Q=(T∗I×T∗SO(3))∪eT∗Qop,whereI=(0,πR)andeT∗QopistherestrictionofthecotangentbundleT∗(Q×Q)ontoQop.ThespaceeT∗QopisthesubmanifoldinT∗Qofthecodimension2;thereforeatypicaltrajectorydoesnotintersectit.Belowweconsideronlysuchtrajectories.ThegroupSO(3)actsbysymplectomorphismsonthesecondfactoroftheproductM:=T∗I×T∗SO(3),endowedwiththestandardsymplecticstructureofacotangentbundle.Therefore,thereducedphasespaceforMhavetheform[27]fM=T∗I×O,2Reducedtwo-bodyproblems3whereOisaSO(3)-orbitw.r.t.thecoadjointactioninthespaceso∗(3)dualtotheLiealgebraso(3).TheorbitOisendowedwiththeKirillovsymplecticform.TheKillingformontheLiealgebraso(3)generatesitsnaturalidentificationwiththedualspaceso∗(3)andmakesboth