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DSMA-TSQM2289FT/UCM/25-92OntheinverseproblemofLagrangiansupermechanicsL.A.Iborty,G.Landiz ,J.Mar n-Solanoy,G.Marmox April25,1997yDpto.deF sicaTe orica.UniversidadComplutense,28040Madrid,Spain.zDpto.diScienzeMatematiche,Universit adiTrieste,Pl.Europa1,I-34100Trieste,Italy.xDpto.diScienzeFisiche,Universit adiNapoli,Mostrad’Oltremare,Pad.19,I-80125Napoli,Italy. INFN,SezionediNapoli,Mostrad’Oltremare,Pad.20,I-80125Napoli,Italy.AbstractTheinverseproblemforLagrangiansupermechanicsisinvesti-gated.Asetofnecessaryandsu cientconditionsforasystemofsecondorderdi erentialequationsinsuperspacetoderivefrom(apos-siblynonregular)superlagrangianfunctionaregiven.Theharmonicsuperoscillatorisrevisitedandafamilyofevenandoddalternativesuperlagrangiansareconstructedforit.Finally,wecommentontheexistenceofrecursionoperators.PACS:11.30.Pb.11IntroductionTheinverseproblemofthecalculusofvariationshasattractedalotofatten-tionovertheyears(forarecentreviewsee[Mo90]).Anelegantdescriptionofnecessaryandsu cientconditionsforasecondorderdi erentialequationtoderivefromaLagrangianfunctionweregivenin[Ba80][He82][Cr81].Theseconditionsencodeinaneatgeometricalwaythewell-knownHelmhotzcondi-tionsfortheexistenceofalocalLagrangianfunction.Asimilarprogrammecanbecarriedoutinthesettingofsupermechanics(orpseudomechanics).Oneofthedi cultiesencounteredintryingtoformulatepreciselytheinverseprobleminthiscontext,istoestablishthecorrectnotionsofsecondorderdi erentialequationsandanintrinsicderivationofEuler-Lagrangeequationsfromasuperlagrangianinsuchawaythattheordinaryconditionscanbeeasilysuperized.Suchageometricalfoundationhasbeenestablishedrecently[Ib92]andinthisgeometricalsettingtheinverseproblemofLagrangiansu-permechanicsacquiresastructuresimilartotheinverseprobleminordinaryLagrangianmechanics.Oneofthemainsourcesofinterestforaddressingtheinverseproblemliesinthealreadywell-knownlinkbetween(super)inte-grabilityandtheexistenceofalternativeLagrangians[Cr83].Newfeaturesemergeinthecontextofsupermechanics[La92]aswewilldiscusslaterandtheexistenceofalternativesuperlagrangiansisnotalwaysrelatedtoare-cursionoperator.Thesimpleexampleoftheharmonicsuperoscillatorisrevisited.Itslinearsupersymmetriesandawholefamilyofalternativesu-perlagrangians,bothevenandodd,aredescribed.2TheInverseProblemofLagrangianSuper-mechanicsLetLbeasuperlagrangiande nedonthetangentsupermanifoldofasuper-spacewithlocalsupercoordinates(qa; ),a=1; ;nand =1; ;m.Thecon gurationsuperspacewithlocalsupercoordinates(qa; )canbethoughttobeanjmdimensionalG-supermanifoldwithsuperalgebraBL,therealGrassmannalgebrawithLgenerators[Ba91].Thesupermanifoldis2de nedbyG-smoothtransitionsuperfunctionsbetweensuperdomains qa= a(q; ) = (q; )(1)andthetangentsupermanifoldistheG-supermanifoldwithlocalsupercoor-dinates(qa;_qa; ;_ )transformingbymeansofthetangentcocycle qa= a(q; );_q0a=@ a@qb_qb+@ a@ _ = (q; );_ 0 =@ @qa_qa+@ @ _ :(2)Inmanyparticularinstancesitisenoughtoconsidercon gurationsuper-spaceswhicharegradedmanifolds(Q;AQ)inthesenseofKostant[Ko77].Insuchcasethecocycle(1)de ningthesupermanifoldcanbedrasticallysimpli ed[Ba79]andtakesthesimplerform qa= a(q) = (q) (3)andthetangentsupermanifold(TQ;TAQ)isagainagradedsupermanifoldde nedbythetangentcocycle[Ib92]: qa= a(q);_q0a=@ a@qb_qb = (q) ;_ 0 = (q)_ +@ @qa_qa :(4)Itisasimple,buttedious,computationtocheckthatthe(1;1)-supertensor eldde nedinlocalsupercoordinatesbyS=dqa @@_qa+d @@_ (5)iswell-de ned.Wecanuseittode netheanalogoustotheCartan1-superformas L=S dL=dqa@L@_qa+d @L@_ =@L@_qadqa+( 1)jLj@L@_ d ;(6)3andtheCartan2-superform L= d L(7)ThesuperenergyELisde nedasthesuperfunctionEL= (L) L=_qa@L@_qa+_ @L@_ L(8)where denotestheLiouvillesuper eldde nedby =_qa@@_qa+_ @@_ .ThesuperlagrangianLissaidtobenondegenerateorregularifthesu-permatrixde nedbythesuperform Linthebasis(dqa;d_qa;d ;d_ )isinvertible.Thisisequivalenttosaythattheweakkernelof L,de nedasthesetofsupervector eldsZsuchthat ( L(Z;X))=0foranyX,iszero.By wemeanthatwetaketheordinarypartofthesuperfunction L(Z;X).IfLisregular,itissimpletoshowthatthedynamicalequationi L=dEL(9)hasauniquesolution whichhappenstobeasupersecondorderdi erentialequation(superSODE)[Ib92],i.e.S( )= ,orlocally =_qa@@qa+_ @@ +fa@@_qa+f @@_ (10)wherefa,f ,evenandoddsuperfunctionsrespectively,arethegeneralizedsuperforces.Locallythe owde nedby canbewrittenasthesetofNewton’slikeequations qa=fa(q;_q; ;_ ); =f (q;_q; ;_ ):(11)Itisanimmediateconsequencethat satis estheequationL L=dL;(12)andthen,thedynamicalequationsaboveareequivalenttotheEuler-LagrangesuperequationsforLddt@L@_qa!=@L@qa;ddt@L@_ !=@L@ :(13)4Statementoftheinverseproblem.GivenasuperSODE ,underwhichconditionsdoesthereexistsasuper-lagrangianfunctionLsuchthat isasolutionofthedynamicalequationde nedbyL,namelyi L=dEL?Proposition1.Anecessaryandsu cientsetofconditionsassuringtheexistenceofa(local)superlagrangianforthesuperSODE are:i)Thereexistsaclosed2-superform .ii)L =0.iii)iV1^V2 =0forallV1,V2verticalsupervector elds.Averticalsupervector eldisasuperve