Invariance_and_first_integrals_of_canonical_Hamiltonian_equations

《Invariance_and_first_integrals_of_canonical_Hamiltonian_equations》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、InvarianceandfirstintegralsofcanonicalHamiltonianequationsVladimirDorodnitsyn∗andRomanKozlov‡∗KeldyshInstituteofAppliedMathematics,RussianAcademyofScience,MiusskayaPl.4,Moscow,125047,Russia;E-mailaddress:dorod@spp.Keldysh.ru‡DepartmentofMathematics,UniversityofBergen,Joha

2、nnesBrunsgate12,5008Bergen,Norway;E-mailaddress:Roman.Kozlov@math.uib.no01.09.2008AbstractInthispaperweconsidertherelationbetweensymmetriesandfirstintegralsofcanonicalHamiltonianequations.Basedonanewlyestablishedidentity(whichisananalogofwellknownNoether’siden-tityforLagr

3、angianapproach),thisapproachprovidesasimpleandclearwaytoconstructfirstintegralswiththehelpofsymmetriesofaHamiltonian.Theapproachisillustratedbyanumberofexamples,includingequationsofthethree-dimensionalKeplermotion.1IntroductionarXiv:0809.1361v1[math-ph]8Sep2008Ithasbeenkn

4、ownsinceE.Noether’sfundamentalworkthatconservationlawsofdifferentialequationsareconnectedwiththeirsymmetryproperties[1].Forconveniencewepresentheresomewell–knownresults(seealso,forexample,[2],[3],[4])forbothLagrangianandHamiltonianapproachestoconservationlaws(firstintegral

5、s).LetusconsiderthefunctionalZL(u)=L(x,u,u1)dx,(1.1)Ωwherex=(x1,x2,...,xm)areindependentvariables,u=(u1,u2,...,un)arede-kk∂ukpendentvariables,u1=(ui),ui=∂xiarefirstorderderivatives,Ωisadomain1inRmandL(x,u,u)isafirstorderLagrangian.Thefunctional(1.1)achieves1itsextremalvalu

6、eswhenu(x)satisfiestheEuler–LagrangeequationsδL∂L∂L=−Di=0,k=1,...,n,(1.2)δuk∂uk∂ukiwhere∂∂∂kkDi=∂xi+ui∂uk+uji∂uk+···,i=1,...,mjaretotaldifferentiationoperators.Hereandbelowweassumesummationoverrepeatedindexes.Notethatequations(1.2)aresecondorderPDEs.WeconsideraLiepointtr

7、ansformationgroupGgeneratedbytheinfinitesimaloperatori∂k∂X=ξ(x,u)+η(x,u)+...,(1.3)∂xi∂ukwheredotsmeananappropriateprolongationoftheoperatoronpartialderiva-tives[5],[6],[7],[8].ThegroupGiscalledavariationalsymmetryofthefunctionalL(u)ifandonlyiftheLagrangiansatisfies[1]iX(L)

8、+LDi(ξ)=0,(1.4)whereXisthefirstprolongation,i.e.theprolongationofthevectorfieldXonthefirstderivativesuk.We