First published in 1998 this unique monograph treats topological, group-theoretic, and geometric problems of ideal hydrodynamics and magneto-hydrodynamics from a unified point of view. It describes the necessary preliminary notions both in hydrodynamics and pure mathematics with numerous examples and figures. This book, now accepted as one of the main references in the field, is accessible to graduates as well as pure and applied mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential geometry. The updated second edition also contains a survey of recent developments in this now-flourishing field of topological and geometric hydrodynamics.
Bibliographic references
Includes bibliographical references and index.
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Contents
In Lieu of a Preface to the Second Edition
Preface
Acknowledgments
Group and Hamiltonian Structures of Fluid Dynamics
Symmetry groups for a rigid body and an ideal fluid
Lie groups, Lie algebras, and adjoint representation
Coadjoint representation of a Lie group
Definition of the coadjoint representation
Dual of the space of plane divergence-free vector fields
The Lie algebra of divergence-free vector fields and its dual in arbitrary dimension
Left-invariant metrics for an arbitrary group
Applications to hydrodynamics
Hamiltonian structure for the Euler equations
Ideal hydrodynamics on Riemannian manifolds
The Euler hydrodynamic equation on manifolds
Dual space to the Lie algebra of divergence-free fields
Inertia operator of an n-dimensional fluid
Proofs of theorems about divergence-free fields
Conservation laws in higher-dimensional hydrodynamics The group setting of ideal magnetohydrodynamics
Equations of magnetohydrodynamics and the Kirchhoff equations
Magnetic extension of any Lie group
Hamiltonian formulation of the Kirchhoff and magnetohydrodynamics equations
Finite-dimensional approximations of the Euler equation
Approximations by vortex systems in the plane
Nonintegrability of four or more point vortices
Hamiltonian vortex approximations in three dimensions
Finite-dimensional approximations of diffeomorphism groups
The Navier
Stokes equation from the group viewpoint
Topology of Steady Fluid Flows
Classification of three-dimensional steady flows
Stationary Euler solutions and Bernoulli functions
Structural theorems
Variational principles for steady solutions
Minimization of the energy
The Dirichlet problem and steady flows
Relation of two variational principles
Semigroup variational principle for two-dimensional steady flows Stability of stationary points on Lie algebras
Stability of planar fluid flows
Stability criteria for steady flows
Wandering solutions of the Euler equation
Linear and exponential stretching of particles
The linearized and shortened Euler equations
The action-angle variables
Spectrum of the shortened equation
The Squire theorem for shear flows
Steady flows with exponential stretching of particles
Analysis of the linearized Euler equation
Inconclusiveness of the stability test for space steady flows
Features of higher-dimensional steady flows
Generalized Beltrami flows
Structure of four-dimensional steady flows
Topology of the vorticity function
Nonexistence of smooth steady flows and sharpness of the restrictions
Topological Properties of Magnetic and Vorticity Fields
Minimal energy and helicity of a frozen-in field
Variational problem for magnetic energy two-dimensional steady flows Extremal fields and their topology
Helicity bounds the energy
Helicity of fields on manifolds
Topological obstructions to energy relaxation
Model example : Two linked flux tubes
Energy lower bound for nontrivial linking
Sakharov-Zeldovich minimization problem
Asymptotic linking number
Asymptotic linking number of a pair of trajectories
Digression on the Gauss formula
Another definition of the asymptotic linking number
Linking forms on manifolds
Asymptotic crossing number
Energy minoration for generic vector fields
Asymptotic crossing number of knots and links
Conformal modulus of a torus
Energy of a knot
Energy of a charged loop
Generalizations of the knot energy
Generalized helicities and linking numbers
Relative helicity
Ergodic meaning of higher-dimensional helicity integrals
Higher-order linking integrals
Calugareanu invariant and self-linking numbeready flows Holomorphic linking number
Asymptotic holonomy and applications
Jones-Witten invariants for vector fields
Interpretation of Godbillon-Vey-type characteristic classes
Differential Geometry of Diffeomorphism Groups
Preliminaries in differential geometry
The Lobachevsky plane of affine transformations
Curvature and parallel translation
Behavior of geodesies on curved manifolds
Relation of the covariant and Lie derivatives
Sectional curvatures of Lie groups
Geometry of the group of torus diffeomorphisms
The curvature tensor for the group of torus diffeomorphisms
Curvature calculations
Diffeomorphism groups and unreliable forecasts
Curvatures of various diffeomorphism groups
Unreliability of long-term weather predictions
Exterior geometry of the group of diffeomorphisms
Conjugate points in diffeomorphism groups^^^er linking integrals
Calugareanu invariant and self-linking numbeready flows Getting around the finiteness of the diameter of the group of volume-preserving diffeomorphisms (by A. Shnirelman)
Interplay between the internal and external geometry of the diffeomorphism group
Diameter of the diffeomorphism groups
Comparison of the metrics and completion of the group of diffeomorphisms
The absence of the shortest path
Discrete flows
Outline of the proofs
Generalized flows
Approximation of generalized flows by smooth ones
Existence of cut and conjugate points on diffeomorphism groups
Infinite diameter of the group of symplectomorphisms
Right-invariant metrics on symplectomorphisms
Calabi invariant
Bi-invariant metrics and pseudometrics on the group of Hamiltonian diffeomorphisms
Bi-invariant indefinite metric and action functional on the group of volume-preserving diffeomorphisms of a three-fold
Kinematic Fast Dynamo Problems
Dynamo and particle stretching
Fast and slow kinematic dynamos
Nondissipative dynamos on arbitrary manifolds
Discrete dynamos in two dimensions
Dynamo from the cat map on a torus
Horseshoes and multiple foldings in dynamo constructions Dissipative dynamos on surfaces
Asymptotic Lefschetz number
Main antidynamo theorems
Cowling's and Zeldovich's theorems
Antidynamo theorems for tensor densities
Digression on the Fokker- Planck equation
Proofs of the antidynamo theorems
Discrete versions of antidynamo theorems
Three-dimensional dynamo models
"Rope dynamo" mechanism
Numerical evidence of the dynamo effect
A dissipative dynamo model on a three-dimensional Riemannian manifold
Geodesic flows and differential operations on surfaces of constant negative curvature
Energy balance and singularities of the Euler equation
Dynamo exponents in terms of topological entropy
Topological entropy of dynamical systems
Bounds for the exponents in nondissipative dynamo models
Upper bounds for dissipative L¹-dynamos
Dynamical Systems with Hydrodynamic Background
The Korteweg-de Vries equation as an Euler equation
Virasoro algebra The translation argument principle and integrability of the higher-dimensional rigid body
Integrability of the KdV equation
Digression on Lie algebra cohomology and the Gelfand -Fuchs cocycle
Equations of gas dynamics and compressible fluids
Barotropic fluids and gas dynamics
Other conservative fluid systems
Infinite conductivity equation
Dynamical systems on the space of knots
Geometric structures on the set of embedded curves
Filament-, Nonlinear Schrödinger-, and Heisenberg chain equations
Loop groups and the general Landau-Lifschitz equation
Sobolev's equation
Elliptic coordinates from the hydrodynamic viewpoint
Charges on quadrics in three dimensions
Charges on higher-dimensional quadrics
References
Appendix (by B. Khesin)
Recent Developments in Topological Hydrodynamics
The hydrodynamic Euler equation as the geodesic flow-- Arnold's framework for the Euler equations
Hamiltonian approach to incompressible fluids
Isovorticed fields, Casimirs, and coadjoint orbits of the group of volumorphisms
Singular vorticities : point vortices and finite-dimensional approximations
Singular vorticities : vortex filaments and membranes
Compressible fluids and semidirect product algebras
Nonuniqueness of weak solutions and the Navier-Stokes equation
Variational principles for groupoids
Structure of steady flows in 3D : Beltrami fields
Generalized Beltrami fields
Steady solutions via symplectic and contact geometry
Eulerian and Lagrangian instability
KAM and near-steady solutions
Helicity and asymptotic linking
Vortex and magnetic reconnections in viscous fluids
Differential Geometry of Diffeomorphism Groupser equation as the geodesic flow-- Otto calculus on the space of densities
Curvatures, conjugate points, and shock waves on diffeomorphism groups
Various metrics on diffeomorphism groups and spaces of densities
Fredholmness of exponential maps on diffeomorphism groups and smoothness of Eider solutions
Kinematic dynamo equations
Dynamo models
Suspension of the cat map
Tokamaks and stellarators
Group and bihamiltonian properties of the KdV, CH, and HS equations
Variations on the Sobolev equation and billiard maps
Symplectic geometry of knots and membranes
Hasimoto and Madelung transforms
Index.
ISBN
9783030742775 (hardbound)
3030742776 (hardbound)
OCLC
1255364498
International Article Number
9783030742775
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