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arXiv:gr-qc/9412041v114Dec1994ENERGY-MOMENTUMCONSERVATIONLAWINHAMILTONIANFIELDTHEORYGennadiASardanashvilyDepartmentofTheoreticalPhysics,PhysicsFaculty,MoscowStateUniversity,117234Moscow,RussiaE-mail:sard@grav.phys.msu.suAbstractIntheLagrangianfieldtheory,onegetsdifferentidentitiesfordifferentstressenergy-momentumtensors,e.g.,canonicalenergy-momentumtensors.Moreover,theseidentitiesarenotconservationlawsoftheabove-mentionedenergy-momentumtensorsingeneral.IntheframeworkofthemultimomentumHamiltonianformalism,wehavethefundamentalidentitywhoserestrictiontoaconstraintspacecanbetreatedtheenergy-momentumconservationlaw.Instandardfieldmodels,thisappearsthemetricenergy-momentumconservationlaw.1IntroductionWefollowthegenerallyacceptedgeometricdescriptionofclassicalfieldsbysectionsoffibredmanifoldsY→X.Theirdynamicsisphrasedintermsofjetspaces(see[2,6,9,12]forthebibliography).GivenafibredmanifoldY→X,thek-orderjetspaceJkYofYcomprisestheequivalenceclassesjkxs,x∈X,ofsectionssofYidentifiedbythefirst(k+1)termsoftheirTaylorseriesatapointx.Oneexploitsthewell-knownfactsthat:(i)thek-jetspaceofsectionsofafibredmanifoldYisafinite-dimensionalsmoothmanifoldand(ii)ak-orderdifferentialoperatoronsectionsofafibredmanifoldYcanbedescribedasamorphismofJkYtoavectorbundleoverX.Asaconsequence,thedynamicsoffieldsystemsisplayedoutonfinite-dimensionalconfigurationandphasespaces.Moreover,thisdynamicsisphrasedinthegeometrictermsduetothe1:1correspondencebetweenthesectionsofthejetbundleJ1Y→YandtheconnectionsonthefibredmanifoldY→X[9,12,13].Infieldtheory,wecanrestrictourselvestothefirstorderLagrangianformalismwhentheconfigurationspaceisJ1Y.Givenfibredcoordinates(xμ,yi)ofY,thejetspaceJ1Yisendowedwiththeadaptedcoordinates(xμ,yi,yiμ).AfirstorderLagrangiandensityontheconfigurationspaceJ1YisrepresentedbyahorizontalexteriordensityL=L(xμ,yi,yiμ)ω,ω=dx1∧...∧dxn,n=dimX.ThecorrespondingfirstorderEuler-LagrangeequationsforsectionssofthefibredjetmanifoldJ1Y→Xread∂λsi=siλ,∂iL−(∂λ+sjλ∂j+∂λsjμ∂μj)∂λiL=0.(1)1Conservationlawsareusuallyrelatedwithsymmetries.ThereareseveralapproachestoexaminesymmetriesintheLagrangianformalism,inparticular,injetterms[3,4,12].WeconsidertheLiederivativesofLagrangiandensitiesinordertoobtaindifferentialconservationlaws.TheseareconservationlawsofNoethercurrentsandtheenergy-momentumconservationlaws.IftheLiederivativeofaLagrangiandensityLalongaverticalvectorfielduGonabundleYisequaltozero,thecurrentconservationlawonsolutionsofthefirstorderEuler-Lagrangeequations(1)takesplace.Inthiscase,theverticalvectorfielduGplaystheroleofageneratorofinternalsymmetries.Incaseoftheenergy-momentumconservationlaw,avectorfieldonYisnotverticalandtheLiederivativeofLdoesnotvanishingeneral.Therefore,thisconservationlawisnotrelatedwithasymmetry.Moreover,onecannotsayaprioriwhatisconserved.Ingravitationtheory,thefirstintegralofgravitationalequationsistheconservationlawofthemetricenergy-momentumtensorofmatter,butonlyinthepresenceofthegrav-itationalfieldgeneratedbythismatter.Inothermodels,themetricenergy-momentumtensorholdsaposteriori.Letu=uμ∂μ+ui∂ibeavectorfieldonafibredmanifoldYanduitsjetliftontothefibredjetmanifoldJ1Y→X.GivenaLagrangiandensityLonJ1Y,letuscomputertheLiederivativeLuL.OnsolutionssofthefirstorderEuler-Lagrangeequations(1),wehavetheequalitys∗LuL=ddxλ[πλi(s)(ui−uμsiμ)+uλL(s)]ω,πμi=∂μiL.(2)Inparticular,ifuisaverticalvectorfieldsuchthatLuL=0,theequality(2)takestheformofthecurrentconservationlawddxλ[uiπλi(s)]=0.(3)Ingaugetheory,thisconservationlawisexemplifiedbytheNoetheridentities.Letτ=τλ∂λbeavectorfieldonXandu=τΓ=τμ(∂μ+Γiμ∂i)itshorizontalliftontothefibredmanifoldYbyaconnectionΓonY.Inthiscase,theequality(2)takestheforms∗LτΓL=−ddxλ[τμTΓλμ(s)]ω(4)2whereTΓλμ(s)=πλi(siμ−Γiμ)−δλμL(5)isthecanonicalenergy-momentumtensorofafieldswithrespecttotheconnectionΓonY.Thetensor(5)istheparticularcaseofthestressenergy-momentumtensors[1,3,5].Notethat,incomparisonwiththecurrentconservationlaws,theLiederivativeLτΓLfailstobeequaltozeroasarule,andtheequality(4)isnottheconservationlawofanycanonicalenergy-momentumtensoringeneral.Itdoesnotlookfundamental,otherwiseitsHamiltoniancounterpart.IntheframerworkofthemultimomentumHamiltonianformalism,wegetthefundamentalidentity(52)whoserestrictiontoaconstraintspacecanbetreatedtheenergy-momentumconservationlaw.Instandardfieldmodels,thisappearsthemetricenergy-momentumconservationlawinthepresenceofabackgroundworldmetric.Lagrangiandensitiesoffieldmodelsarealmostalwaysdegenerateandthecorrespond-ingEuler-Lagrangeequationsareunderdetermined.Todescribeconstraintfieldsystems,themultimomentumHamiltonianformalismcanbeutilized[8,10,11].Intheframeworkofthisformalism,thephinite-dimensionalphasespaceoffieldsistheLegendrebundleΠ=n∧T∗X⊗YTX⊗YV∗Y(6)overYintowhichtheLegendremorphismbLassociatedwithaLagrangiandensityLonJ1Ytakesitsvalues.Thisphasespaceisprovidedwiththefibredcoordinates(xλ,yi,pλi)suchthat(xμ,yi,pμi)◦bL=(xμ,yi,πμi).TheLegendrebundle(6)carriesthemultisymplecticformΩ=dpλi∧dyi∧ω⊗∂λ.(7)IncaseofX=R,theformΩrecoversthestandardsymplecticforminanalyticalme-chanics.BuildingonthemultisymplecticformΩ,onecandevel
本文标题:Energy-Momentum Conservation Laws in Hamiltonian F
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