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Arnold V.I., Khesin B.A. — Topological methods in hydrodynamics
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Название: Topological methods in hydrodynamics
Авторы: Arnold V.I., Khesin B.A.
Аннотация: Topological hydrodynamics is a young branch of mathematics studying topological features of flows with complicated trajectories, as well as their applications to fluid motions. It is situated at the crossroad of hyrdodynamical stability theory, Riemannian and symplectic geometry, magnetohydrodynamics, theory of Lie algebras and Lie groups, knot theory, and dynamical systems. Applications of this approach include topological classification of steady fluid flows, descriptions of the Korteweg-de Vries equation as a geodesic flow, and results on Riemannian geometry of diffeomorphism groups, explaining, in particular, why longterm dynamical weather forecasts are not reliable. Topological Methods in Hydrodynamics is the first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics for a unified point of view. The necessary preliminary notions both in hydrodynamics and pure mathematics are described with plenty of examples and figures. The book is accessible to graduate students as well as to both pure and applied mathematicians working in the fields of hydrodynamics, Lie groups, dynamical systems and differential geometry.
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Рубрика: Математика /Математическая Физика /
Статус предметного указателя: Готов указатель с номерами страниц
ed2k: ed2k stats
Год издания: 1998
Количество страниц: 391
Добавлена в каталог: 23.04.2005
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Предметный указатель
-plane equation 323
ABC flows 72—73 286—287
Action along a path 231
Action along a path, least action principle 1 16
Action-angle variables 100 308
Adjoint operator 4
Adjoint operator, orbit 8
Adjoint operator, representation 4
Aharonov — Bohm effect 199
Almost complex structure 330
Angular momentum, relative to the body 16
Angular momentum, relative to the space (or spatial) 16
Angular velocity 4 15
Angular velocity, relative to the body 15
Angular velocity, spatial 15
Anosov map 265
Antidynamo theorems 273—277
Antidynamo theorems, discrete 284—285
Antidynamo theorems, proofs of 281—284
Approximation, of generalized flows 238
Approximation, of vortex equation 56—59
Asymptotic, crossing number 152—155
Asymptotic, linking number 140—144
Attractor 64 66
Attractor, dimension of attractor 66
Attractor, general 279
Barotropic fluid 318
Beltrami fields 72
Beltrami fields, generalized 110
Bernoulli function 70 110
Bernoulli function, sequence 268
Bernoulli function, surface 111
Berry phase 58 199
Betti number 276
Bi-Hamiltonian system 309
Bi-invariant (pseudo-)metrics 252
Biot — Savart integral 146
Borromean rings 132(fig) 174(fig) 175
Brownian motion 134
Burgers equation 306
Calabi invariant 246—249
Calabi invariant, form 249
Calabi invariant, integral 249
Calugareanu formula 178
Cascade of solitori 168(fig)
Casimir functions 47 48 312
Cat map 104(fig) 265(fig) 266 288
Characteristic path length 202 218
Chern — Simons functional 189
Circulations of vortices 57
Clebsch variables 324
Coadjoint representation 10—11
Coadjoint representation, invariants of 47—48
Coadjoint representation, orbit 11
Coboundary 304 316
Cocycle 304 316
Cohomology complex 316
Cohomology complex, group 316
Commutant subalgebra 35 257
Commutant subalgebra, subgroup 35 247
Commutator in Lie algebra 7
Commutator in Lie algebra, of vector fields 7 13
Compatible Poisson structures 309
Completely integrable system 308
Complex structure 330
Configuration space of fluid 32
Confocal quadrics 340 342
Conformal modulus 159
Conjugate point 224
Conjugate point, first 224 224(fig) 241
Conservation laws 43—44 55 62
Coriolis-type term 323
Coset of 1-forms
Cosymmetry 114
Covariant derivative 119 202
Cowling theorem 274
Cross-helicity 55
Crossing number 152
Crossing number, asymptotic 152 153 155
Crossing number, average 153
Crossing number, of a knot 155
Crossing number, topological 154
Curvature tensor 200
Curvature, constant negative 197 293
Curvature, Ricci 200
Curvature, scalar 201
Curvature, sectional 200
Cut point 240—241
De Rham current 59 186 328
de Rham current, linking form 147—151
Degree, of a function 157
Degree, of a map 144
Degree, of a satellite link 155
Density form 320
Derivative, covariant 199 202
Derivative, exterior 32
Derivative, inner 32
Derivative, Lie 5—6 32—33
Derivative, Schwarzian 317
Derivative, variational 291 313
Diameter problem 226
Diffeomorphism, attainable 227—228
Diffeomorphism, Hamiltonian 242
Diffeomorphism, structurally stable 338
Diffeomorphism, symplectic 242
Diffeomorphism, volume-preserving 2
Diffeomorphism, volume-preserving, exact 14 35
Dirichlet integral 78
Dirichlet integral, problem 336
Dirichlet integral, variational problem 78—80
Discrete flow 234—235
Displacement energy 253
Divergence 125
Dual space of a Lie algebra 10 33—34
Dynamo equation 259
Dynamo equation, kinematic 259
Dynamo equation, self-consistent 262
Dynamo, discrete 261
Dynamo, dissipative 261
Dynamo, fast 260—261
Dynamo, in -norm 264
Dynamo, kinematic 260
Dynamo, mean 270
Dynamo, nondissipative 261
Dynamo, rope 286
Dynamo, self-consistent 262
Dynamo, slow 260
Elliptic coordinates 340 342
Energy 2
Energy, -energy 153
Energy, Hamiltonian function 37
Energy, minimization 75
Energy, of a curve 160
Energy, of a vector field 19 75 119
Energy, of MHD system 50
Energy, relaxation 129
Energy, second differential of 86
Energy, spectrum of a knot 162
Energy-Casimir method 95
Energy-momentum method 95—96
Enstrophy invariants 43
Entrophy 299—300
Equation in variations 88
Euler equation, of ideal hydrodynamics 20
Euler equation, of ideal hydrodynamics, aproximation of 56—59
Euler equation, of ideal hydrodynamics, generalized 37
Euler equation, of ideal hydrodynamics, on Riemannian manifolds 31
Euler equation, of ideal hydrodynamics, stationary 69—70
Euler equation, of rigid body 16—17
Euler theorem, first 16
Euler theorem, second 16
Euler — Helmholtz equation 20 56
Exact diffeomorphism 14 35
Existence and uniqueness theorems in hydrodynamics 24—25
Exponential instability 100 102 216
Extension of a Lie algebra 304
Exterior derivative 32
Extra symmetries 113 114
Extremal fields 76 80 120—121
Fast dynamo 260—261
Filament equation 332—333
First cut point 240—241
First cut point, local 241
First integral 43—44 308
Fluid, barotropic (or isentropic) 318
Fluid, compressible 322—323
Fluid, incompressible 2
Fluid, perfectly conducting 323
Fluid, viscous 63—67
Fluid, with a free boundary 323
Flux 130 159
Focal quadrics 343
Fokker — Planck equation 277—281
Force-free field 71
Form-potential 125
Form-residue 180
Frozen-in (or transported) field 22—23 49
Galerkin approximation 63 66
Gas dynamics 318 333
Gauss linking number 130
Gauss linking number, theorem (or formula) 143
Gauss — Bonnet theorem 296
Gauss — Codazzi formula 221
Gauss — de Rham linking form 147—151
Gaussian integral 186
Gelfand — Fuchs cocycle 304 317—318
Generalized flow (GF) 236—238
Genus of a knot 156
Geodesic 198
Geodesic, flow 293
Geodesic, submanifold 210
Geodesic, variation 201
Gibbs solution 278
Godbillon — Vey class 191—192
Group 1
Group, Lie group 1
Group, of diffeomorphisms 2
Group, of symplectomorphisms 242
Group, unimodular 97 206
Hamiltonian function 25
Hamiltonian function, field 25 27
Hamiltonian function, KdV structure 313—314
Hamiltonian vortex approximations 56—59
Hasimoto transformation 332
Heisenberg magnetic chain 333
Helicity 43 121 125
Helicity, invariance theorem 122 (see also “Hopf invariant”)
Helicity, relative 167
Helicity, theorem 141
Helmholtz (or vortex) equation 20 38 47 56 99
Hermitian inner product 108
Hessian matrix 114 223
Hofer's metric 252
Holonomy, asymptotic 187
Holonomy, average 287
Holonomy, functional 190
Homeoidal density 341
Homoclinic point 267 268(fig)
Homothetic quadrics 340
Homotopy formula 33 39 275
Hopf invariant of a field (or helicity) 43 121 125
Hopf invariant of a map 127 128(fig) 174
Hopf vector field 123 141
Horocycle 294
Horocyclic flows 294
Horseshoe 267 268(fig)
Immersed knot 330—331 331
Incompressible fluid 2
Inertia operator 14 36—37
Infinite conductivity equation 324—325
Inner automorphism 3
Inner product, of a vector field with a form (or inner derivative) 12 32
Inner product, on vector fields 36
Inscribed angle theorem 338
Invariant of coadjoint action 47—48
Isentropic fluid see “Barotropic fluid”
Isovorticed fields 80
Ivory theorem 341
Jacobi equation 201
Jacobi equation, identity 7 26
Jones — Witten functional complex 189
Jones — Witten functional complex, for a link 185
Jones — Witten functional complex, for a vector field 188
Kadomtsev — Petviashvili equation 323
Kahler geometry 326
KAM theory 101 133
KdV equation see “Korteweg — de Vries equation”
KdV Hamiltonian structure, first 314
KdV Hamiltonian structure, second 313
Kinematic dynamo 260
kinetic energy 2 16 29
Kirchhoff equations 50
Kirillov — Kostant structure 25 329
Knot invariants 183 326
Knot invariants, Jones — Witten 185
Knot invariants, Vassiliev 183
Kolmogorov's conjectures 64
Korteweg — de Vries equation 305
Kronecker symbol 43 46
Lagrangian coordinates 49
Lagrangian coordinates, instability 216
Lagrangian coordinates, submanifold 253
Landau — Lifschitz equation 333—334
Laplace — Beltrami operator 123 275
Lefschetz formula 273
Lefschetz formula, number 273
Lefschetz formula, number, asymptotic 273
Left-invariant metric 14 19 196
Leibniz identity 26 40
Lenard scheme 310 310(fig) 314
Leray residue 180
Lichnerowicz cocycle 326
Lie algebra 4 7 32
Lie algebra, abstract Lie algebra 7
Lie algebra, dual space of 10
Lie algebra, of vector fields 32
Lie algebra, vector space of 4
Lie bracket of vector fields 32
Lie derivative 6 32 39
Lie group 1 7 32
Lie — Poisson structure (or bracket) 27 310
Lifting indices 37 322
Linear Poisson bracket see “Lie — Poisson bracket”
Linearized vortex equation 99
Linearized vortex equation, Navier — Stokes equation 296
Link invariant, complex 184
Link invariant, Jones — Witten 185
Link invariant, Vassiliev 183
Link, essential 132
Link, satellite 155
Link, trivial 132
Linking form 147—151
Linking form, complex 184
Linking number 130
Linking number, asymptotic 140(fig) 141 144 171
Linking number, average 141 170—171
Linking number, for cascades 167
Linking number, holomorphic 181
Linking number, mutual (or multilinking number) 171—172
Linking number, of third order 175
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