On the existence of collisionless equivariant minimizers for the classical n-body problem.pdf

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1、OntheExistenceofCollisionlessEquivariantMinimizersfortheClassicaln-bodyProblemDavideL.Ferrario∗andSusannaTerracini†February7,2008AbstractWeshowthattheminimizationoftheLagrangianactionfunctionalonsuit-ableclassesofsymmetricloopsyieldscollisionlessperiodicorbitsoft

2、hen-bodyproblem,providedthatsomesimpleconditionsonthesymmetrygrouparesat-isfied.Moreprecisely,wegiveafairlygeneralconditiononsymmetrygroupsGoftheloopspaceΛforthen-bodyproblem(withpotentialofhomogeneousdegree−α,withα>0)whichensuresthattherestrictionoftheLagrangiana

3、ctionAtothespaceΛGofG-equivariantloopsiscoerciveanditsminimizersarecollisionless,withoutanystrongforceassumption.Manyofthealreadyknownperiodicorbitscanbeprovedtoexistbythisresult,andseveralneworbitsarefoundwithsomeappropriatechoiceofG.MSCSubj.Class:Primary70F10(M

4、echanicsofparticlesandsystems:n-bodyproblems);Secondary70F16(Mechanicsofparticlesandsystems:Collisionsincelestialmechanics,regularization),37C80(Dynamicalsystemsandergodictheory:Symmetries,equivariantdynamicalsystems),70G75(Mechanicsofparticlesandsystems:Variatio

5、nalmethods).Keywords:symmetricperiodicorbits,n-bodyproblem,collisions,minimizersoftheLagrangianactionarXiv:math-ph/0302022v39May20031IntroductionThemethodofminimizingtheLagrangianactiononaspaceofloopssymmetricwithrespecttoawell-chosensymmetrygrouphasbeenusedinsom

6、erecentpaperstofindnewinterestingperiodicorbitsforthen-bodyproblem[15,14,7,23].Suchavariationalapproachhasbeenextensivelyexploitedinthelastdecadesbyseveralotherauthors.Wereferthereadertothefollowingarticlesandreferencestherein:[2,4,5,6,11,10,17,22,25,26,27,30,31,3

7、2,33,36].Thisapproachconsistsinseekingperiodictrajectoriesascriticalpointsoftheactionfunctionalassociated∗DipartimentodiMatematicadelPolitecnicodiMilano,PiazzaLeonardodaVinci,32;20133Milano,Italy.email:ferrario@mate.polimi.it†DipartimentodiMatematicaeApplicazioni

8、UniversitdegliStudidiMilano-BicoccaViaBicoccadegliArcimboldi,8;20126Milano,Italy.email:suster@matapp.unimib.it1toasystemofnparticleswithmassesmi>0,interactingt