Topological Methods in Hydrodynamics

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Springer Science & Business Media, Apr 13, 1998 - Computers - 374 pages
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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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Contents

lowledgments
1
The group setting of ideal magnetohydrodynamics
49
12 The NavierStokes equation from the group viewpoint
63
4 Asymptotic linking number
139
B Asymptotic crossing number of knots and links
155
7 Generalized helicities and linking numbers
166
8 Asymptotic holonomy and applications
184
Differential Geometry of Diffeomorphism Groups
195
Kinematic Fast Dynamo Problems 259
258
constructions
267
5 Dynamo exponents in terms of topological entropy
299
chain equations
331
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About the author (1998)

Arnol'd, Steklov Mathematical Institute

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