Topological Methods in Hydrodynamics

In Lieu of a Preface to the Second Edition
Preface
Acknowledgments
Contents
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 magnetohydrodynamics
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 three dimensions
11.D Finite-dimensional approximations of diffeomorphism groups
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.F 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 Sakharov–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 on 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 torus
6 Energy of a knot
6.A Energy of a charged loop
6.B Generalizations of the knot energy
7 Generalized helicities and linking numbers
7.A Relative helicity
7.B Ergodic meaning of higher-dimensional helicity integrals
7.C Higher-order linking integrals
7.D Calugareanu 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 invariant metric
3 Riemannian geometry of the group of area-preserving diffeomorphisms of the two-torus
3.A The curvature tensor for the group of torus diffeomorphisms
3.B Curvature calculations
4 Diffeomorphism groups and unreliable forecasts
4.A Curvatures of various diffeomorphism groups
4.B Unreliability of long-term weather predictions
5 Exterior geometry of the group of volume-preserving diffeomorphisms
6 Conjugate points in diffeomorphism groups
7 Getting around the finiteness of the diameter of the group of volume-preserving diffeomorphisms (by A. Shnirelman)
7.A Interplay between the internal and external geometry of the diffeomorphism group
7.B Diameter of the diffeomorphism groups
7.C Comparison of the metrics and completion of the group of diffeomorphisms
7.D The absence of the shortest path
7.E Discrete flows
7.F Outline of the proofs
7.G Generalized flows
7.H Approximation of generalized flows by smooth ones
7.I Existence of cut and conjugate points on diffeomorphism groups
8 Infinite diameter of the group of Hamiltonian diffeomorphisms and symplecto-hydrodynamics
8.A Right-invariant metrics on symplectomorphisms
8.B Calabi invariant
8.C Bi-invariant metrics and pseudometrics on the group of Hamiltonian diffeomorphisms
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 particle stretching
1.A Fast and slow kinematic dynamos
1.B Nondissipative dynamos on arbitrary manifolds
2 Discrete dynamos 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 antidynamo theorems
3.A Cowling's and Zeldovich's theorems
3.B Antidynamo theorems for tensor densities
3.C Digression on the Fokker–Planck equation
3.D Proofs of the antidynamo theorems
3.E Discrete versions of antidynamo theorems
4 Three-dimensional dynamo models
4.A "Rope dynamo" mechanism
4.B Numerical evidence of the dynamo effect
4.C A dissipative dynamo model on a three-dimensional Riemannian manifold
4.D Geodesic ows and differential operations on surfaces of constant negative curvature
4.E Energy balance and singularities of the Euler equation
5 Dynamo exponents in terms of topological entropy
5.A Topological entropy of dynamical systems
5.B Bounds for the exponents in nondissipative dynamo models
5.C Upper bounds for dissipative L^1-dynamos
VI Dynamical Systems with Hydrodynamic Background
1 The Korteweg–de Vries equation as an Euler equation
1.A Virasoro algebra
1.B The translation argument principle and integrability of the higher-dimensional rigid body
1.C Integrability of the KdV equation
1.D Digression on Lie algebra cohomology and the Gelfand–Fuchs cocycle
2 Equations of gas dynamics and compressible fluids
2.A Barotropic fluids and gas dynamics
2.B Other conservative
fluid systems
2.C Infinite conductivity equation
3 Kahler geometry and dynamical systems on the space of knots
3.A Geometric structures on the set of embedded curves
3.B Filament-, Nonlinear Schrödinger-, and Heisenberg chain equations
3.C Loop groups and the general Landau–Lifschitz equation
4 Sobolev's equation
5 Elliptic coordinates from the hydrodynamic viewpoint
5.A Charges on quadrics in three dimensions
5.B Charges on higher-dimensional quadrics
References
Recent Developments in Topological Hydrodynamics
A. I Group and Hamiltonian Structures of Fluid Dynamics
I.1 The hydrodynamic Euler equation as the geodesic flow
I.2 Arnold's framework for the Euler equations
I.3 Hamiltonian approach to incompressible fluids
I.4 Isovorticed fields, Casimirs, and coadjoint orbits of the group of volumorphisms
I.5 Singular vorticities: point vortices and finite-dimensional approximations
I.6 Singular vorticities: vortex filaments and membranes
I.7 Compressible fluids and semidirect product algebras
I.8 Nonuniqueness of weak solutions and the Navier–Stokes equation
I.9 Variational principles for groupoids
A. II Topology of Steady Fluid Flows
II.1 Structure of steady flows in 3D: Beltrami fields
II.2 Generalized Beltrami fields
II.3 Steady solutions via symplectic and contact geometry
II.4 Eulerian and Lagrangian instability
II.5 KAM and near-steady solutions
A. III Topological Properties of Magnetic and Vorticity Fields
III.1 Helicity and asymptotic linking
III.2 Vortex and magnetic reconnections in viscous fluids
A. IV Differential Geometry of Diffeomorphism Groups
IV.1 Otto calculus on the space of densities
IV.2 Curvatures, conjugate points, and shock waves on diffeomorphism groups
IV.3 Various metrics on diffeomorphism groups and spaces of densities
IV.4 Fredholmness of exponential maps on diffeomorphism groups and smoothness of Euler solutions
A.V Kinematic Fast Dynamo Problems
V.1 Kinematic dynamo equations
V.2 Dynamo models
V.3 Suspension of the cat map
V.4 Tokamaks and stellarators
A. VI Dynamical Systems with Hydrodynamic Background
VI.1 Group and bihamiltonian properties of the KdV, CH, and HS equations
VI.2 Variations on the Sobolev equation and billiard maps
VI.3 Symplectic geometry of knots and membranes
VI.4 Hasimoto and Madelung transforms
References
Index