An alternative form of the Helmholtz criterion in the inverse problem of calculus of variat

We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent va

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aANALTERNATIVEFORMOFTHEHELMHOLTZCRITERIONINTHEINVERSEPROBLEMOFCALCULUSOFVARIATIONSTHEODOREVORONOVToAlanWeinsteinontheoccasionofhis60thbirthdayAbstract.Wegiveanecessaryandsu cientconditionfortheexistenceofalocalsolutionoftheinverseproblemofcalculusofvariationsintermsoftheidenticalvanishingofthevariationofafunctionalonanextendedspace(withthenumberofindepen-dentvariablesincreasedbyone),andexplainitsrelationwiththeclassicalHelmholtzcriterionusingthedeRhamcomplexonanin nite-dimensionalspaceof elds.1.IntroductionThispaperdealswiththequestion,underwhichconditionsgivenfunctionscanbethevariationalderivativesofsomefunctional.(Theprecisesetupisdescribedin§2.)Thisisaclassicalquestion,andthereisaclassicalanswertoitgivenbytheso-calledHelmholtz(orHelmholtz–Volterra)criterion.Onecansee[7]foranexpositionandahistoricalreview.TheHelmholtzconditionis,infact,nothingbuttheclosednessofa1-formofaparticularappearanceonanin nite-dimensional“spaceof elds”onwhichfunctionalsunderconsiderationarede ned,thoughthismaybehiddeninsomeexpositions.HencetheHelmholtzcriterioncanbeviewedasaversionofanin nite-dimensionalPoincar´elemmafor1-forms,andprovedaccordingly.Intheordinary nite-dimensionalcaseweknowthata1-formisclosedifitsintegralsoverpathsdonotchangeundersmallperturbationofapath xingtheboundary.Thisisalsotruefork-formsandk-paths.Pushedtothelimit,thisideaallowstoexpresstheusualdeRham

di erentialentirelyinvariationalterms.(Forsupermanifoldsthisis,actually,theonlywayonecanconstructanadequateanalogoftheCartan–deRhamcomplex,see[11,8,9],also[4],[1],[6].)Moreover,thedeRhamcomplexisembeddedinalargercomplexconsistingofallLagrangiansofk-pathsonamanifold[10].

Inthisnotewesuggestanalternativecriterionoftheexistenceofalocalsolutionoftheinverseproblemofcalculusofvariationsinterms