Second-order multisymplectic field theory A variat

arXiv:math/9909100v1[math.DG]17Sep1999AVARIATIONALAPPROACHTOSECOND-ORDERMULTISYMPLECTICFIELDTHEORYSHINARKOURANBAEVAANDSTEVESHKOLLERAbstract.Thispaperpresentsageometric-variationalapproachtocontin-uousanddiscretesecond-orderfieldtheoriesfollowingthemethodologyof[MPS1998].StayingentirelyintheLagrangianframeworkandlettingYde-notetheconfigurationfiberbundle,weshowthatboththemultisymplecticstructureonJ3YaswellastheNoethertheoremarisefromthefirstvaria-tionoftheactionfunction.Wegeneralizethemultisymplecticformformuladerivedforfirstorderfieldtheoriesin[MPS1998],tothecaseofsecond-orderfieldtheories,andweapplyourtheorytotheCamassa-Holm(CH)equationinboththecontinuousanddiscretesettings.Ourdiscretizationproducesamultisymplectic-momentumintegrator,ageneralizationoftheMoser-VeselovrigidbodyalgorithmtothesettingofnonlinearPDEswithsecondorderLa-grangians.Contents1.Introduction22.VariationalPrinciplesforsecondorderClassicalFieldTheory32.1.Multisymplecticgeometry32.2.Thevariationalroutetothemultisymplecticform.42.3.Themultisymplecticformformula132.4.TheNoetherTheorem153.AmultisymplecticapproachtotheCamassa-Holmequation163.1.TheCamassa-Holmequation163.2.ThemultisymplecticformformulafortheCHequation184.Discretesecond-ordermultisymplecticfieldtheory204.1.Ageneralconstruction204.2.Amultisymplectic-momentumalgorithmfortheCHequation224.3.ThediscreteEuler-Lagrangeequations.244.4.ThediscreteCartanform.264.5.Thediscretemultisymplecticformformula.264.6.ThediscreteNoethertheorem.28Acknowledgments31References31Date:September1,1998;currentversionMarch29,1999.Keywordsandphrases.Multisymplecticgeometry,shallowwaterequations.12S.KOURANBAEVAANDS.SHKOLLER1.IntroductionThispapercontinuesthedevelopmentofthevariationalapproachtomultisym-plecticfieldtheoryintroducedinMarsden,Patrick,andShkoller[MPS1998].Inthatpaper,onlyfirst-orderfieldtheorieswereconsidered.Herein,weshallfocusonsecond-orderfieldtheories,i.e.,thosefieldtheoriesgovernedbyLagrangiansthatdependonthespacetimelocation,thefield,anditsfirstandsecondpartialderivatives.Multisymplecticgeometryanditsapplicationstocovariantfieldtheoryandnon-linearpartialdifferentialequations(PDE)hasarichandinterestinghistorythatweshallnotdiscussinthispaper;rather,wereferthereaderto[Ga1974,G1991,GIM1999,MPS1998]andthereferencestherein.Thecovariantmultisymplecticapproachisthefield-theoreticgeneralizationofthesymplecticapproachtoclassi-calmechanics.TheconfigurationmanifoldQofclassicalLagrangianmechanicsisreplacedbyafiberbundleY→Xoverthen+1dimensionalspacetimemanifoldX,whosesectionsarethephysicalfieldsofinterest;theLagrangianphasespaceisTQinLagrangianmechanics,whereasforkth-orderfieldtheories,theroleofphasespaceisplayedbythekth-jetbundleofY,JkY,thusreflectingtheadditionaldependenceofthefieldsonspatialvariables.ForagivensmoothLagrangianL:TQ→R,thereisadistinguishedsym-plectic2-formωLonTQ,whoseHamiltonianvectorfieldisthesolutionoftheEuler-LagrangeequationsofLagrangianmechanics.Lagrangianfieldtheories,ontheotherhand,governedbycovariantLagrangiansL:JkY→Λn+1(X),canbecompletelydescribedbythemultisymplecticn+2-formΩLonJ2k−1Y,thefield-theoreticanalogueofthesymplectic2-formωLofclassicalmechanics.InthecasethatXisonedimensional,ΩLreducestotheusualtime-dependent2-formofclas-sicalnonautonomousmechanics(see[MS1999]).Traditionally,thesymplectic2-formωLaswellasthemultisymplecticn+2-formΩLareconstructedontheLagrangianside,usingthepull-backbytheLegendretransformofcanonicaldifferentialformsonthedualorHamiltonianside.Recently,however,Marsden,Patrick,andShkoller[MPS1998]haveshownthatforfirstorderfieldtheorieswhereinL:J1Y→Λn+1(X),ΩL=dΘLarisesastheboundaryterminthefirstvariationoftheactionRXL◦j1φforsmoothmappingsφ:X→Y.Thismethodisadvantageoustothetraditionalapproachinthat(i)acompletegeometrictheorycanbederivedwhilestayingentirelyontheLagrangianside,and(ii)multisymplecticstructurecanbeobtainedinnon-standardsettingssuchasdiscretefieldtheory.Thepurposeofthispaper,istogeneralizetheresultsof[MPS1998]tothecasethatL:J2Y→Λn+1(X).InSection2,weproveinTheorem2.1,thatauniquemultisymplecticn+2formarisesastheboundarytermofthefirstvariationoftheactionfunction.WethenproveinTheorem2.2,themultisymplecticformformulaforsecond-orderfieldtheories,acovariantgeneralizationofthefactthatinconservativemechanics,theflowpreservesthesymplecticstructure.WethenobtainthecovariantNoethertheoremforsecond-orderfieldtheories,bytakingthefirstvariationoftheactionfunction,restrictedtothespaceofsolutionsofthecovariantEuler-Lagrangeequations.InSection3,weuseourabstractgeometrictheoryontheCamassa-Holm(CH)equation,amodelofshallowwaterwavesthatsimultaneouslyexhibitssolitarySECOND-ORDERMULTISYMPLECTICFIELDTHEORY3waveinteractionandwave-breaking.Weshowthatthemultisymplecticformfor-mulaproducesanewconservationlawideallysuitedtostudywaveinstability,andconnectourintrinsictheorywithBridges’theoryofmultisymplecticstructures(see[B1997]and[MPS1998]).Section4isdevotedtothediscretizationofsecond-orderfieldtheories.WeareabletouseourgeneraltheorytoproducenumericalalgorithmsfornonlinearPDEthataregovernedbysecond-orderLagrangianswhic