流体动力学中的拓扑方法（英文版）

图书标准信息

• 作者 [法]阿诺德 著
• 出版社 世界图书出版公司
• 出版时间 2009-08
• 版次 1
• ISBN 9787510005305
• 定价 45.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 376页
• 正文语种 英语

【内容简介】

Hydrodynamicsisoneofthosefundamentalareasinmathematicswhereprogressatanymomentmayberegardedasastandardtomeasuretherealsuccessofmath-meticalscience.Manyimportantachievementsinthisfieldarebasedonprofoundtheoriesratherthanonexperiments.Inram，thosehydrodynamicaltheoriesstimulateddevelopmentsinthedomainsofpuremathematics，suchascomplexanalysis，topology，stabilitytheory，bifurcationtheory，andcompletelyintegraldynamicalsystems.Inspiteofallthisacknowledgedsuccess，hydrodynamicswithitsspec-tabularempiricallawsremainsachallengeformathematicians.

【目录】

Preface
Acknowledgments
I.GroupandHamiltonianStructuresofFluidDynamics
1.Symmetrygroupsforarigidbodyandanidealfluid
2.Liegroups，Liealgebras，andadjointrepresentation
3.CoadjointrepresentationofaLiegroup
3.A.Definitionofthecoadjointrepresentation
3.B.Dualofthespaceofplanedivergence-freevectorfields
3.C.TheLiealgebraofdivergence-freevectorfieldsanditsdualinarbitrarydimension
4.Left-invariantmetricsandarigidbodyforanarbitrarygroup
5.Applicationstohydrodynamics
6.HamiltonianstructurefortheEulerequations
7.IdealhydrodynamicsonRiemannianmanifolds
7.A.TheEulerhydrodynamicequationonmanifolds
7.B.DualspacetotheLiealgebraofdivergence-freefields
7.C.Inertiaoperatorofann-dimensionalfluid
8.ProofsoftheoremsabouttheLiealgebraofdivergence-freefieldsanditsdual
9.Conservationlawsinhigher-dimensionalhydrodynamics
10.Thegroupsettingofidealmagnetohydrodyuamics
10.A.EquationsofmagnetohydrodynamicsandtheKirchhoffequations
10.B.MagneticextensionofanyLiegroup
10.C.HamiltonianformulationoftheKirchhoffandmagnetohydrodynamicsequations
11.Finite-dimensionalapproximationsoftheEulerequation
11.A.Approximationsbyvortexsystemsintheplane
11.B.Nonintegrabilityoffourormorepointvortices
11.C.Hamiltonianvortexapproximationsinthreedimensions
11.D.Finite-dimensionalapproximationsofdiffeomorphismgroups
12.TheNavier-Stokesequationfromthegroupviewpoint

II.TopologyofSteadyFluidFlows
1.Classificationofthree-dimensionalsteadyflows
1.A.StationaryEulersolutionsandBernoullifunctions
1.B.Structuraltheorems
2.Variationalprinciplesforsteadysolutionsandapplicationstotwo-dimensionalflows
2.A.Minimizationoftheenergy
2.B.TheDirichletproblemandsteadyflows
2.C.Relationoftwovariationalprinciples
2.D.Semigroupvariationalprinciplefortwo-dimensionalsteadyflows
3.StabilityofstationarypointsonLiealgebras
4.Stabilityofplanarfluidflows
4.A.Stabilitycriteriaforsteadyflows
4.B.WanderingsolutionsoftheEulerequation
5.Linearandexponentialstretchingofparticlesandrapidly
oscillatingperturbations
5.A.ThelinearizedandshortenedEulerequations
5.B.Theaction-anglevariables
5.C.Spectrumoftheshortenedequation
5.D.TheSquiretheoremforshearflows
5.E.Steadyflowswithexponentialstretchingofparticles
5.EAnalysisofthelinearizedEulerequation
5.G.Inconclusivenessofthestabilitytestforspacesteadyflows
6.Featuresofhigher-dimensionalsteadyflows
6.A.GeneralizedBeltramiflows
6.B.Structureoffour-dimensionalsteadyflows
6.C.Topologyofthevorticityfunction
6.D.Nonexistenceofsmoothsteadyflowsandsharpnessof
therestrictions

III.TopologicalPropertiesofMagneticandVorticityFields
1.Minimalenergyandhelicityofafrozen-infield
1.A.Variationalproblemformagneticenergy
1.B.Extremalfieldsandtheirtopology
1.C.Helicityboundstheenergy
1.D.Helicityoffieldsonmanifolds
2.Topologicalobstructionstoenergyrelaxation
2.A.Modelexample：Twolinkedfluxtubes
2.B.Energylowerboundfornontriviallinking
3.Salcharov-Zeldovichminimizationproblem
4.Asymptoticlinkingnumber
4.A.Asymptoticlinkingnumberofapairoftrajectories
4.B.DigressionontheGaussformula
4.C.Anotherdefinitionoftheasymptoticlinkingnumber
4.D.Linkingformsosmanifolds
5.Asymptoticcrossingnumber
5.A.Energyminorationforgenericvectorfields
5.B.Asymptoticcrossingnumberofknotsandlinks
5.C.Conformalmodulusofatoras
6.Energyofaknot
6.A.Energyofachargedloop
6.B.Generalizationsoftheknotenergy
7.Generalizedbelicitiesandlinkingnumbers
7.A.Relativebelicity
7.B.Ergodicmeaningofhigher-dimensionalhelicityintegrals
7.C.Higher-orderlinkingintegrals
7.D.Cahigareanuinvariantandself-linkingnumber
7.E.Holomorphiclinkingnumber
8.Asymptoticholonomyandapplications
8.A.Jones-Witteninvariantsforvectorfields
8.B.InterpretationofGodbillon-Vey-typecharacteristicclasses

ⅣDifferentialGeometryofDiffeomorphismGroups
1.TheLobachevskyplaneandpreliminariesindifferentialgeometry
1.A.TheLobachevskyplaneofaffinetransformations
1.B.Curvatureandparalleltranslation
1.C.Behaviorofgeodesicsoncurvedmanifolds
1.D.RelationofthecovariantandLiederivatives
2.SectionalcurvaturesofLiegroupsequippedwithaone-sidedinvadantmetric
3.Riemanniangeometryofthegroupofarea-preservingdiffeomorphismsofthetwo-torus
3.A.Thecurvaturetensorforthegroupoftomsdiffeomorphisms
3.B.Curvaturecalculations
4.Diffeomorphismgroupsandunreliableforecasts
4.A.Curvaturesofvarionsdiffeomorqhismgroups
4.B.Unreliabilityoflong-termweatherpredictions
5.Exteriorgeometryofthegroupofvolume-preservingdiffeomophisms
6.Conjugatepointsindiffeomorphismgroups
4.Asymptoticlinkingnumber
4.A.Asymptoticlinkingnumberofapairoftrajectories
4.B.DigressionontheGaussformula
4.C.Anotherdefinitionoftheasymptoticlinkingnumber
4.D.Linkingformsosmanifolds
5.Asymptoticcrossingnumber
5.A.Energyminorationforgenericvectorfields
5.B.Asymptoticcrossingnumberofknotsandlinks
5.C.Conformalmodulusofatoras
6.Energyofaknot
6.A.Energyofachargedloop
6.B.Generalizationsoftheknotenergy
7.Generalizedbelicitiesandlinkingnumbers
7.A.Relativebelicity
7.B.Ergodicmeaningofhigher-dimensionalhelicityintegrals
7.C.Higher-orderlinkingintegrals
7.D.Cahigareanuinvariantandself-linkingnumber
7.E.Holomorphiclinkingnumber
8.Asymptoticholonomyandapplications
8.A.Jones-Witteninvariantsforvectorfields
8.B.InterpretationofGodbillon-Vey-typecharacteristicclasses
8.C.Bi-invariantmetricsandpseudometricsonthegroupofHamiltoniandiffeomotphisms
8.D.Bi-invariantindefinitemetricandactionfunctionalonthegroupofvolume-preservingdiffeomorphismsofathree-fold

V.KinematicFastDynamoProblems
1.Dynamoandpaniclestretching
1.A.Fastandslowkinematicdynamos
1.B.Nondissipativedynamosonarbitrarymanifolds
2.Discretedymmosintwodimensions
2.A.Dynamofromthecatmaponatorus
2.B.Horseshoesandmultiplefoldingsindynamo
constructions
2.C.Dissipativedynamosonsurfaces
2.D.AsymptoticLefschetznumber
3.Mainant/dynamotheorems
3.A.CowfingsandZeldovichstheorems
3.B.Antidynamotheoremsfortensordensities
3.C.DigreasionontheFokker-Planekequation
3.D.Proofsoftheantidynamotheorems
3.E.Discreteversionsofantidynemotheorems
4.Threc-dimensionaldynamomodels
4.A.“Ropedynamo”mechanism
4.B.Numericalevidenceofthedynamoeffect

VI.DynamicalSystemsWithHydrodynamicalBackgroud
References
Index