流体动力学中的拓扑方法

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版次：1
页数：376
字数：
印刷时间：2009年08月01日
开本：24开
纸张：胶版纸
包装：平装
是否套装：否
国际标准书号ISBN：9787510005305
作者:（法）阿诺德著出版社:世界图书出版公司出版时间:2009年08月

目　　录

Preface
Acknowledgments
I.Group and Hamiltonian Structures of Fluid Dynamics
1.Symmetry groups for a rigid body and an ideal fluid
2.Lie groups, Lie algebras, and adjoint representation
3.Coadjoint representation of a Lie group
3.A.Definition of the coadjoint representation
3.B.Dual of the space of plane divergence-free vector fields
3.C.The Lie algebra of divergence-free vector fields and its
dual in arbitrary dimension
4.Left-invariant metrics and a rigid body for an arbitrary group
5.Applications to hydrodynamics
6.Hamiltonian structure for the Euler equations
7.Ideal hydrodynamics on Riemannian manifolds

Preface Acknowledgments I.Group and Hamiltonian Structures of Fluid Dynamics 1.Symmetry groups for a rigid body and an ideal fluid 2.Lie groups, Lie algebras, and adjoint representation 3.Coadjoint representation of a Lie group 3.A.Definition of the coadjoint representation 3.B.Dual of the space of plane divergence-free vector fields 3.C.The Lie algebra of divergence-free vector fields and its dual in arbitrary dimension 4.Left-invariant metrics and a rigid body for an arbitrary group 5.Applications to hydrodynamics 6.Hamiltonian structure for the Euler equations 7.Ideal hydrodynamics on Riemannian manifolds 7.A.The Euler hydrodynamic equation on manifolds 7.B.Dual space to the Lie algebra of divergence-free fields 7.C.Inertia operator of an n-dimensional fluid 8.Proofs of theorems about the Lie algebra of divergence-free fields and its dual 9.Conservation laws in higher-dimensional hydrodynamics 10.The group setting of ideal magnetohydrodyuamics 10.A.Equations of magnetohydrodynamics and the Kirchhoff equations 10.B.Magnetic extension of any Lie group 10.C.Hamiltonian formulation of the Kirchhoff and magnetohydrodynamics equations 11.Finite-dimensional approximations of the Euler equation 11.A.Approximations by vortex systems in the plane 11.B.Nonintegrability of four or more point vortices 11.C.Hamiltonian vortex approximations in threedimensions 11.D.Finite-dimensional approximations of diffeomorphismgroups 12.The Navier-Stokes equation from the group viewpoint II.Topology of Steady Fluid Flows 1.Classification of three-dimensional steady flows 1.A.Stationary Euler solutions and Bernoulli functions 1.B.Structural theorems 2.Variational principles for steady solutions and applications to two-dimensional flows 2.A.Minimization of the energy 2.B.The Dirichlet problem and steady flows 2.C.Relation of two variational principles 2.D.Semigroup variational principle for two-dimensional steady flows 3.Stability of stationary points on Lie algebras 4.Stability of planar fluid flows 4.A.Stability criteria for steady flows 4.B.Wandering solutions of the Euler equation 5.Linear and exponential stretching of particles and rapidly oscillating perturbations 5.A.The linearized and shortened Euler equations 5.B.The action-angle variables 5.C.Spectrum of the shortened equation 5.D.The Squire theorem for shear flows 5.E.Steady flows with exponential stretching of particles 5.E Analysis of the linearized Euler equation 5.G.Inconclusiveness of the stability test for space steady flows 6.Features of higher-dimensional steady flows 6.A.Generalized Beltrami flows 6.B.Structure of four-dimensional steady flows 6.C.Topology of the vorticity function 6.D.Nonexistence of smooth steady flows and sharpness of the restrictions III.Topological Properties of Magnetic and Vorticity Fields 1.Minimal energy and helicity of a frozen-in field 1.A.Variational problem for magnetic energy I.B.Extremal fields and their topology 1.C.Helicity bounds the energy 1.D.Helicity of fields on manifolds 2.Topological obstructions to energy relaxation 2.A.Model example: Two linked flux tubes 2.B.Energy lower bound for nontrivial linking 3.Salcharov-Zeldovich minimization problem …… IV.Differential Geometry of diffeomorphism Groups V.Kinematic Fast Dynarno Problems VI.Dynamical Systems With Hydrodynamical Backgroud References Index

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