An Index Theorem for Non Periodic Solutions of Ham

arXiv:math/9911047v4[math.DG]30Oct2000ANINDEXTHEOREMFORNONPERIODICSOLUTIONSOFHAMILTONIANSYSTEMSPAOLOPICCIONEANDDANIELV.TAUSKABSTRACT.WeconsideraHamiltoniansetup(M,ω,H,L,Γ,P),where(M,ω)isasymplecticmanifold,LisadistributionofLagrangiansubspacesinM,PaLagrangiansubmanifoldofM,HisasmoothtimedependentHamiltonianfunctiononMandΓ:[a,b]→MisanintegralcurveoftheHamiltonianflow~HstartingatP.WedonotrequireanyconvexitypropertyoftheHamiltonianfunctionH.Undertheassump-tionthatΓ(b)isnotP-focalitisintroducedtheMaslovindeximaslov(Γ)ofΓgivenintermsofthefirstrelativehomologygroupoftheLagrangianGrassmannian;undergenericcircumstancesimaslov(Γ)iscomputedasasortofalgebraiccountoftheP-focalpointsalongΓ.WeprovethefollowingversionoftheIndexTheorem:undersuitablehypothe-ses,theMorseindexoftheLagrangianactionfunctionalrestrictedtosuitablevariationsofΓisequaltothesumofimaslov(Γ)andaconvexitytermoftheHamiltonianHrela-tivetothesubmanifoldP.WhentheresultisappliedtothecaseofthecotangentbundleM=TM∗ofasemi-Riemannianmanifold(M,g)andtothegeodesicHamiltonianH(q,p)=12g−1(p,p),weobtainasemi-RiemannianversionofthecelebratedMorseIndexTheoremforgeodesicswithvariableendpointsinRiemanniangeometry.1.INTRODUCTIONOurinterestintheindextheoryforsolutionsoftheHamiltonequationsinasymplec-ticmanifoldwasoriginallymotivatedbytheaimofextendingtothecaseofnonpositivedefinitemetricstheclassicalresultsoftheMorsetheoryforgeodesicsinRiemannianman-ifolds(see[20]).Despitethisoriginalmotivation,thegeometricapplicationsofthetheorydevelopedarelefttotheverylastpartofthearticle,andmostoftheresultspresentedinthepaperbelongindeedtotherealmofthetheoryofsystemsofordinarydifferentialequationswithcoefficientsintheLiealgebrasp(2n,IR)ofthesymplecticgroup.Suchsystemswillbecalledsymplecticdifferentialsystems.Inordertomotivatethetheorypresentedinthispaper,wegiveashortaccountofthemathematicalhistoryoftheproblem.TheoriginoftheindextheoryistobefoundintheSturmtheoryforordinarydifferentialequations(seeforinstance[5]).TheSturmoscilla-tiontheoremdealswithsecondorderdifferentialequationsoftheform−(px′)′+rx=λxwherepandrarefunctionswithp0,andλisarealparameter.ThetheoremstatesthatthenumberofzeroesofanonnullsolutionxoftheSturmequationsatisfyingx(a)=0equalstheindexoftheindexformI(x1,x2)=Rba[px′1x′2+rx1x2]dtdefinedinspaceofrealvaluedmapson[a,b]vanishingattheendpoints.TheextensionoftheresultsoftheSturmiantheorytothecaseofsystemsofdifferentialequationsisessentiallyduetoMorse,obtainingthecelebratedMorseIndexTheoreminRiemanniangeometry(seeforinstance[6,20]).TheMorse–SturmsystemsforwhichtheDate:revisedversionofJune2000.2000MathematicsSubjectClassification.37J05,53C22,53C50,53D12,70H05.ThefirstauthorispartiallysponsoredbyCNPq,thesecondauthorissponsoredbyFAPESP.1ANINDEXTHEOREMFORHAMILTONIANSYSTEMS2theoremappliesarethoseoftheformg−1(gv′)′=Rv,whereg(t)isapositivedefinitesymmetricmatrixandR(t)isg(t)-symmetriclinearoperatoronIRnforallt.Suchsys-temsareobtained,forinstance,byconsideringtheJacobiequationalongageodesicinaRiemannianmanifold;theequationisconvertedintoasystemofODE’sinIRnbymeansofaparalleltrivializationofthetangentbundleofthemanifoldalongthegeodesic.Inthissituation,theindexformI(v,w)=Rbag(v′,w′)+g(Rv,w)dthasfiniteindexinthespaceofvectorvaluedfunctionon[a,b]vanishingattheendpoints,anditisequaltothenumberofconjugatepointsofthesystemintheinterval]a,b[.Byminorchanges,theMorseIndexTheoremisalsovalidinthecaseofLorentzianmetricsg,i.e.,metricshavingindexone,providedthatoneconsiderscausal,i.e.,timelikeorlightlike,geodesicsandthatonerestrictstheindexformItovectorfieldsthatarepointwiseorthogonaltothegeodesic(see[3]).SubsequentresultshaveextendedtheIndexTheoremtothecaseofsolutionswithnonfixedinitialand/orfinalendpoint(see[15,23])andtothecaseofordinarydifferentialoperatorsofevenorder(see[9]).WhenpassingtothecaseofspacelikegeodesicsinLorentzianmanifolds,or,moreingeneral,togeodesicsinsemi-Riemannianmanifoldsendowedwithmetrictensorsofarbitraryindex,thereisnohopetoextendtheoriginalformulationoftheMorseIndexTheoremforseveralreasons.Infirstplace,theactionfunctionalisstronglyindefinite,i.e.,itssecondvariation,givenbytheindexformI,hasalwaysinfiniteindex.Moreover,thesetofconjugatepointsalongagivengeodesicmayfailtobediscrete,andtheJacobidifferentialoperatorisnolongerself-adjoint.Adifferentintegervaluedgeometricinvariant,calledtheMaslovindex,hasbeenre-centlyintroducedinthecontextofsemi-Riemanniangeodesics(see[11,13,19]).InthecaseofRiemannianorcausalLorentziangeodesics,theMaslovindexcoincideswiththegeometricindexofthegeodesic,whichisthenumberofconjugatepointscountedwithmultiplicity.Formetricsofarbitraryindex,undergenericcircumstances,thisindexiscomputedasasortofalgebraiccountoftheconjugatepointsalongthegeodesic.Itisnaturaltoexpectthat,asinthecaseofpositivedefinitemetrics(see[8]),theMaslovindexshouldplaytheroleofthegeometricindexinmetricswitharbitraryindex,andinthispaperwepresentseveralargumentstostrengthenthisidea.Namely,weshowthattheMaslovindex,undercertaincircumstances,isequaltotheindexoftherestrictionofItoasuitablesubspaceofvariationalve