【理想流体的哈密尔描述】hamilton description of ideal fluid

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1、Hamiltoniandescriptionoftheideal¯uid*†P.J.MorrisonDepartmentofPhysicsandInstituteforFusionStudies,UniversityofTexas,Austin,Texas78712-1060TheHamiltonianviewpointof¯uidmechanicalsystemswithfewandin®nitenumberofdegreesoffreedomisdescribed.Rudimentaryconceptsof®nite-degree-of-freedomHamiltoniandynamics

2、arereviewed,inthecontextofthepassiveadvectionofascalarortracer®eldbya¯uid.Thenotionsofintegrability,invariant-tori,chaos,overlapcriteria,andinvariant-toribreakuparedescribedinthiscontext.Preparatorytotheintroductionof®eldtheories,systemswithanin®nitenumberofdegreesoffreedom,elementsoffunctionalcalcu

3、lusandactionprinciplesofmechanicsarereviewed.Theactionprinciplefortheidealcompressible¯uidisdescribedintermsofLagrangianormaterialvariables.Hamiltoniansystemsintermsofnoncanonicalvariablesarepresented,includingseveralexamplesofEulerianorinviscid¯uiddynamics.Liegrouptheorysuf®cientforthetreatmentofre

4、ductionisreviewed.ThereductionfromLagrangiantoEulerianvariablesistreatedalongwithClebschvariabledecompositions.StabilityinthecanonicalandnoncanonicalHamiltoniancontextsisdescribed.Suf®cientconditionsforstability,suchasRayleigh-likecriteria,areseentobeonlysuf®cientinthegeneralcasebecauseoftheexistenc

5、eofnegative-energymodes,whicharepossessedbyinteresting¯uidequilibria.Linearlystableequilibriawithnegativeenergymodesarearguedtobeunstablewhennonlinearityordissipationisadded.Theenergy-Casimirmethodisdiscussedandavariantofitthatdependsuponthenotionofdynamicalaccessibilityisdescribed.Theenergycontento

6、faperturbationaboutageneral¯uidequilibriumiscalculatedusingthreemethods.[S0034-6861(98)00102-0]CONTENTSV.TutorialonLieGroupsandAlgebras,Reduction,andClebschVariables491A.TutorialonLiegroupsandLiealgebras491I.Introduction4681.Liegroups492II.RudimentsofHamiltonianSystemswithFew2.Liealgebras493Degreeso

7、fFreedom,Illustratedby3.Realizationandrepresentation495PassiveAdvectioninTwo-DimensionalFluids469B.Reduction496A.Amodelfortwo-dimensional¯uidmotion4691.Reductionof®nite-dimensionalsystems496B.Passivea