关注微信

推荐商品

    加载中... 正在为您读取数据...
分享到:
  • 流体动力学中的拓扑方法(英文版)[平装]
  • 共2个商家     36.00元~36.00
  • 作者:阿诺德(作者)
  • 出版社:世界图书出版公司;第1版(2009年8月1日)
  • 出版时间:
  • 版次 :
  • 印刷时间:
  • 包装:
  • ISBN:9787510005305

  • 商家报价
  • 简介
  • 评价
  • 加载中... 正在为您读取数据...
  • 商品描述

    编辑推荐

    《流体动力学中的拓扑方法(英文版)》是由世界图书出版公司出版的。

    作者简介

    作者:(法国)阿诺德

    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
    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
    1.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
    4. Asymptotic linking number
    4.A.Asymptotic linking number of a pair of trajectories
    4.B.Digression on the Gauss formula
    4.C.Another definition of the asymptotic linking number
    4.D.Linking forms os manifolds
    5. Asymptotic crossing number
    5.A.Energy minoration for generic vector fields
    5.B.Asymptotic crossing number of knots and links
    5.C.Conformal modulus of a toras
    6. Energy of a knot
    6.A.Energy of a charged loop
    6.B.Generalizations of the knot energy
    7. Generalized belicities and linking numbers
    7.A.Relative belicity
    7.B.Ergodic meaning of higher-dimensional helicity integrals
    7.C.Higher-order linking integrals
    7.D.Cahigareanu invariant and self-linking number
    7.E.Holomorphic linking number
    8. Asymptotic holonomy and applications
    8.A.Jones-Witten invariants for vector fields
    8.B.Interpretation of Godbillon-Vey-type characteristic classes

    Ⅳ Differential Geometry of Diffeomorphism Groups
    1. The Lobachevsky plane and preliminaries in differential geometry
    1.A.The Lobachevsky plane of affine transformations
    1.B.Curvature and parallel translation
    1.C.Behavior of geodesics on curved manifolds
    1.D.Relation of the covariant and Lie derivatives
    2. Sectional curvatures of Lie groups equipped with a one-sided invadant metric
    3. Riemannian geometry of the group of area-preserving diffeomorphisms of the two-torus
    3.A.The curvature tensor for the group of toms diffeomorphisms
    3.B.Curvature calculations
    4. Diffeomorphism groups and unreliable forecasts
    4.A.Curvatures of varions diffeomorqhism groups
    4.B.Unreliability of long-term weather predictions
    5. Exterior geometry of the group of volume-preserving diffeomophisms
    6. Conjugate points in diffeomorphism groups
    4. Asymptotic linking number
    4.A.Asymptotic linking number of a pair of trajectories
    4.B.Digression on the Gauss formula
    4.C.Another definition of the asymptotic linking number
    4.D.Linking forms os manifolds
    5. Asymptotic crossing number
    5.A.Energy minoration for generic vector fields
    5.B.Asymptotic crossing number of knots and links
    5.C.Conformal modulus of a toras
    6. Energy of a knot
    6.A.Energy of a charged loop
    6.B.Generalizations of the knot energy
    7. Generalized belicities and linking numbers
    7.A.Relative belicity
    7.B.Ergodic meaning of higher-dimensional helicity integrals
    7.C.Higher-order linking integrals
    7.D.Cahigareanu invariant and self-linking number
    7.E.Holomorphic linking number
    8. Asymptotic holonomy and applications
    8.A.Jones-Witten invariants for vector fields
    8.B.Interpretation of Godbillon-Vey-type characteristic classes
    8.C. Bi-invariant metrics and pseudometrics on the groupof Hamiltonian diffeomotphisms
    8.D. Bi-invariant indefinite metric and action functional on the group of volume-preserving diffeomorphisms of a three-fold

    V. Kinematic Fast Dynamo Problems
    1. Dynamo and panicle stretching
    1.A. Fast and slow kinematic dynamos
    1.B. Nondissipative dynamos on arbitrary manifolds
    2. Discrete dymmos in two dimensions
    2.A. Dynamo from the cat map on a torus
    2.B.Horseshoes and multiple foldings in dynamo
    constructions
    2.C.Dissipative dynamos on surfaces
    2.D.Asymptotic Lefschetz number
    3. Main ant/dynamo theorems
    3.A.Cowfing's and Zeldovich's theorems
    3.B.Antidynamo theorems for tensor densities
    3.C.Digreasion on the Fokker-Planek equation
    3.D.Proofs of the antidynamo theorems
    3.E.Discrete versions of antidynemo theorems
    4. Threc-dimensional dynamo models
    4.A. “Rope dynamo” mechanism
    4.B.Numerical evidence of the dynamo effect

    VI.Dynamical Systems With Hydrodynamical Backgroud
    References
    Index

    序言

    Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of math-metical science. Many important achievements in this field are based on profound theories rather than on experiments. In ram, those hydro dynamical theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology, stability theory, bifurcation theory, and completely integral dynamical systems. In spite of all this acknowledged success, hydrodynamics with its spec-tabular empirical laws remains a challenge for mathematicians. For instance, the phenomenon of turbulence has not yet acquired a rigorous mathematical theory. Furthermore, the existence problems for the smooth solutions of hydrodynamic equations of a three-dimensional fluid are still open. The simplest but already very substantial mathematical model for fluid dynamics is the hydrodynamics of an ideal (i.e., of an incompressible and in viscid)homogeneous fluid. From the mathematical point of view.

    文摘

    插图: