On the k-Symplectic, k-Cosymplectic and Multisympl

arXiv:0705.4364v1[math-ph]30May2007ONTHEk-SYMPLECTIC,k-COSYMPLECTICANDMULTISYMPLECTICFORMALISMSOFCLASSICALFIELDTHEORIESNarcisoRom´an-Roy∗DepartamentodeMatem´aticaAplicadaIV.EdificioC-3,CampusNorteUPCC/JordiGirona1.08034Barcelona.Spain´AngelM.Rey†,ModestoSalgado‡,SilviaVilari˜no§DepartamentodeXeometr´ıaeTopolox´ıaFacultadedeMatem´aticas,UniversidadedeSantiagodeCompostela,15706-SantiagodeCompostela,SpainFebruary1,2008AbstractTheobjectiveofthisworkistwofold:First,weanalyzetherelationbetweenthek-cosymplecticandthek-symplecticHamiltonianandLagrangianformalismsinclassicalfieldtheories.Inparticular,weprovetheequivalencebetweenk-symplecticfieldtheoriesandtheso-calledautonomousk-cosymplecticfieldtheories,extendinginthiswaythedescriptionofthesymplecticformalismofautonomoussystemsasaparticularcaseofthecosymplecticformalisminnon-autonomousmechanics.Furthermore,weclarifysomeaspectsofthegeo-metriccharacterofthesolutionstotheHamilton-deDonder-WeylandtheEuler-Lagrangeequationsintheseformalisms.Second,westudytheequivalencebetweenk-cosymplecticandaparticularkindofmultisymplecticHamiltonianandLagrangianfieldtheories(thosewheretheconfigurationbundleofthetheoryistrivial).Keywords:k-symplecticmanifolds,k-cosymplecticmanifolds,multisymplecticmanifolds,HamiltonianandLagrangianfieldtheories.AMSs.c.(2000):70S05,53D05,53D10∗e-mail:[e-mail]†e-mail:[e-mail]‡e-mail:[e-mail]§e-mail:[e-mail]´an-Royetal,k-symplectic,k-cosymplecticandmultisymplecticformalisms...2Contents1Introduction22k-symplecticandk-cosymplecticHamiltonianformalisms32.1k-vectorfieldsandintegralsections..........................32.2k-symplecticandk-cosymplecticmanifolds......................42.3k-symplecticHamiltoniansystems..........................62.4k-cosymplecticHamiltoniansystems.........................82.5Autonomousk-cosymplecticHamiltoniansystems..................93k-symplecticandk-cosymplecticLagrangianformalisms113.1CanonicalstructuresinthebundlesT1kQandRk×T1kQ..............113.2k-symplecticLagrangianformalism..........................123.3k-cosymplecticLagrangianformalismandautonomousk-cosymplecticLagrangiansystems.........................................144MultisymplecticHamiltonianformalism164.1Multisymplecticmanifoldsandmultimomentumbundles..............164.2MultisymplecticHamiltonianformalism.......................174.3Relationwiththek-cosymplecticHamiltonianformalism..............185MultisymplecticLagrangianformalism215.1MultisymplecticLagrangiansystems.........................215.2Relationbetweenmultisymplecticandk-cosymplecticLagrangiansystems....221IntroductionThek-symplecticandk-cosymplecticformalismsarethesimplestgeometricframeworksfordescribingclassicalfieldtheories.Thek-symplecticformalism[13,25](alsocalledpolysymplec-ticformalism)isthegeneralizationtofieldtheoriesofthestandardsymplecticformalisminautonomousmechanics,andisusedtogiveageometricdescriptionofcertainkindsoffieldthe-ories:inalocaldescription,thosewhoseLagrangianandHamiltonianfunctionsdonotdependonthecoordinatesinthebasis(inmanyofthem,thespace-timecoordinates).Thefounda-tionsofthek-symplecticformalismarethek-symplecticmanifoldsintoducedin[2,3,4].Thek-cosymplecticformalismisthegeneralizationtofieldtheoriesofthestandardcosymplecticformalismfornon-autonomousmechanics,[21,22],anditdescribesfieldtheoriesinvolvingthecoordinatesinthebasisontheLagrangianandontheHamiltonian.Thefoundationsofthek-cosymplecticformalismarethek-cosymplecticmanifoldsintroducedin[21,22].Oneoftheadvantagesoftheseformalismsisthatonlythetangentandcotangentbundleofamanifoldarerequiredfortheirdevelopment.(Abriefreviewofk-symplecticandk-cosymplecticgeometryN.Rom´an-Royetal,k-symplectic,k-cosymplecticandmultisymplecticformalisms...3isgiveninSection2.2).Otherdifferentpolysymplecticformalismsfordescribingfieldtheorieshavebeenproposedin[10,11,15,23,26,27,30].Intheseformalisms,thefieldequations(Hamilton-deDonder-WeylandEuler-Lagrangeequa-tions)canbewritteninageometricalwayusingintegrablek-vectorfields.However,althoughintegralsectionsofintegrablek-vectorfields(i.e.,integrabledistributions)thataresolutionstothegeometricalfieldequationsareprovedtobesolutionstotheHamilton-deDonder-WeylortheEuler-Lagrangeequations,theconverseisnotalwaystrue.Thisalsooccurswhenothergeometricdescriptionsofclassicalfieldtheoriesintermsofmultivectorfieldsareconsidered(see[7,8,28]fordetailsinthecaseofmultisymplecticfieldtheories).Hereweprovethat,inthek-cosymplecticformalism,everysolutiontotheHamilton-deDonder-Weylequationsis,infact,anintegralsectionofanintegrablek-vectorfieldthatisasolutiontothegeometricalfieldequa-tionsintheHamiltonianformalism.Nevertheless,inthek-symplecticHamiltonianformalism,thisisnolongertrue,unlesssomeadditionalconditionsonthesolutionstotheHamilton-deDonder-Weylarerequired.AllthesefeaturesarediscussedinSections2.3,2.4,2.5,3.2,and3.3.Afterreviewingthek-cosymplecticHamiltonianformalisminSection2.4,Section2.5containsotherrelevantresultsofthiswork.Inparticular,therelationbetweenthek-cosymplecticandthek-symplecticHamiltoni