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Books, Free Shipping, Topological Methods In Hydrodynamics

Posted at: Offers to Sell and Export | Posted on: Tue 27 Jul, 2010 7:59 am | Product Category: Reference books [3760]
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Product Details

Author Arnold

Publisher World Publishing Corporation

Isbn 9787510005305, 7510005302


August 2009

Pages 376

Size 24k

Edition First

Format Paperback

Language English

Page material gelatine plate paper

Product Description:

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.




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



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