Topological Methods in Hydrodynamics

✍ Scribed by Vladimir I. Arnold, Boris A. Khesin (auth.)

• Publisher: Springer-Verlag New York
• Year: 1998
• Tongue: English
• Leaves: 392
• Series: Applied Mathematical Sciences 125
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Group and Hamiltonian Structures of Fluid Dynamics....Pages 1-67
Topology of Steady Fluid Flows....Pages 69-118
Topological Properties of Magnetic and Vorticity Fields....Pages 119-193
Differential Geometry of Diffeomorphism Groups....Pages 195-257
Kinematic Fast Dynamo Problems....Pages 259-301
Dynamical Systems with Hydrodynamical Background....Pages 303-344

✦ Subjects

Mathematics, general; Fluids; Complexity; Numerical and Computational Methods in Engineering