流体动力学中的拓扑方法

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.Cowfings and Zeldovichs 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