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Arnold V.I., Khesin B.A. — Topological methods in hydrodynamics
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Название: Topological methods in hydrodynamicsАвторы: 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: Рубрика: Математика /Математическая Физика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1998Количество страниц: 391Добавлена в каталог: 23.04.2005Операции: Положить на полку |
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Предметный указатель
Liouville theorem 308 Lobachevsky plane 197 293—294 Lobachevsky plane, absolute 293 Lobachevsky plane, geodesic flow 293 Local invariants 47 Local invariants, densities 48 Locally flat (or Euclidean) manifold 216 Loop group 334 Loop group, Lie algebra 334 Lorentz force 50 120 Lyapunov, exponent 267 Lyapunov, stable point 84 Magnetic diffusivity 259 Magnetic diffusivity, Reynolds number 259 Magnetic extension 50—52 Magnetic extension, chain equation 333 Magnetohydrodynamics (MHD) equations 49 120 Magri bracket 313 Manakov method 309 Marsden — Weinstein symplectic structure 326—327 Massey product 175 Maurer — Cartan formula 315 Maximum principle 283 Metrics, comparison of 226 228 Metrics, Hofer's 252 Metrics, left-invariant 14 19 196 Metrics, right-invariant 19 MHD equations 49 120 Minimal element 83 Minimizer 76 78—80 Moebius transformation 163 Momentum 15—16 50 Monge — Ampere equation 223 Morse function 114 Morse index 114 Morse index, type of orbit 114 Moser's lemma 136 Moyal product 62 Multiflow 237 Multilinking number 171 Navier — Stokes equation 63 Neumann problem 136 Newlander — Nirenberg theorem 330 Nonlinear Schrodinger equation 332 Null-homologous fields 123 125 Orbit, adjoint 8 81 Orbit, coadjoint 11 81 Orbit, elliptic 133 Orbit, hyperbolic 134 Over-crossing number 155 Pairing of dual spaces 33—34 Parallel translation 198—199 198(fig) Perfect eddy 215 Poincare duality 142 Poincare recurrence theorem 96—97 Poisson bracket, constant 311 Poisson bracket, linear (or Lie — Poisson) 27 310 Poisson bracket, of fields 5 Poisson bracket, of functions 26 28 Poisson bracket, of hydrodynamic type 323 Poisson, Lie — Poisson structure 27 310 Poisson, manifold 26 Poisson, pair 309 Poisson, structure 26 Poisson, structure, compatible 309 Polymorphism 82 Polyvector 22 Principle of least action 1 16 Pseudometric 252 Quadrics, confocal 340 342 Quadrics, confocal, nomothetic 340 Quasiclassical asymptotics 277 280 Rayleigh theorem 92 Regular point of foliation 85 Representation, adjoint 4 Representation, coadjoint 10—11 Reynolds number 64 Reynolds number, magnetic 259 Ricci curvature 200 Riemannian manifold 33 Riemannian manifold, metric on a group 15 Riemannian manifold, metric on a group, left-invariant 15 196 Right-invariant metric 19 Rigid body 1 16 63 Rigid body, in a fluid 50 323 Rigid body, with a cavity 323 Rope dynamo 286 Rotation class 36 Routh method 95 Ruelle — Takens conjecture 66 Sakharov — Zeldovich problem 134 Satellite link 155 Schrodinger equation, nonlinear 332 Schwarzian derivative 317 Second differential (or variation) of the energy 86 90 107 Second fundamental form 220 Seifert surface 156 328 328(fig) Self-linking number 141 168 Semianalytic subset 74 Semidirect product 50 53 319 Semigroup 82 Shallow water equation 307 Shock wave 306 306(fig) Short paths, system of 145 Shortened vortex equation 99 Sine-algebra 60 62 323 Skew gradient 25 Skew-symmetric product 25 Skew-symmetric product, form 26 Smale horseshoe 267 268(fig) Sobolev equation 335 Sobolev equation, inequality 154 Space average 146 Squire theorem 102 Stability in linear approximation 86 Stability theorem 91 Stability theorem, second 91 Stable point 84 Stationary (or steady) flow 69 Stationary (or steady) flow, Euler equation 69—70 Stationary (or steady) flow, form 275 Strange Attractor 64—65 Stream function 9 12 43 56 210 Stretch-twist-fold mechanism 286 286(fig) Symplectic leaf 28 Symplectic leaf, manifold 25 242 Symplectic leaf, manifold, exact 243 Symplectic leaf, structure 25 Symplectic leaf, variables 324 Symplectic leaf, variables, canonical 324 System of short paths 145 Threshold of a potential 280 280(fig) Time average 147 Topological entropy 299—300 Tradewind current 215 215(fig) 218 Translation, left 3 Translation, of the argument 309 311 Translation, right 2 Tunneling coefficients 278 Turbulence 64—67 Twist number 178 178(fig) Two-cocycle 304 Variational derivative 39 313 Variational derivative, problem for energy 75 78 Vassiliev invariant 183 Vector field, divergence-free 4 Vector field, divergence-free, exact 35—36 38 Vector field, Hamiltonian 25 27 248 Vector field, left-(or right-)invariant 7 Vector field, modeled on a link 132 Vector field, modeled on a link, strongly 132 Vector field, null-homologous 123 125 Vector field, of geodesic variation 201 Vector field, semiexact 36 38 Vector momentum 50 Vector-potential 125 146 Virasoro algebra 304 Virasoro algebra, coadjoint orbits 307 314 Virasoro algebra, group 214 304 Vortex 56 Vortex equation 20 56 99 Vortex equation, linearized 99 Vortex equation, shortened 99 Vortex, three-vortex problem 58 Vorticity field 14 22 46 56 Vorticity field, form 22 46 Vorticity field, function 22 46 56 111 Wandering point 269 Wave vector 101 210 Weierstrass example 230 230(fig) Whitehead link 132(fig) Writhing number 178 178(fig) Zeldovich theorem 274 290 Zorn Lemma 83
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