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Arnold V.I., Khesin B.A. — Topological methods in hydrodynamics
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.


Язык: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\beta$-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 $L_d$-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, $E_{3/2}$-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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