This book treats topological, group-theoretic and geometric problems of ideal hydrodynamics and magnetohydrodynamics from a unified point of view. The modern approach is based on the use of hydrodynamical stability theory, Riemannian and symplectic geometry, theory of Lie algebras and Lie groups, knot theory, and dynamical systems. The power of abstract techniques is demonstrated by giving a wide range of applications to such areas as topological classification of steady fluid flows, descriptions of the Korteweg-de Vries equation as a geodesic flow and resulting diffeomorphism groups on Riemannian geometry, explaining in particular, why long-term dynamical weather forecasts are not reliable. The monograph is addressed to graduate students as well as to pure and applied mathematicians working in the fields of hydrodynamics, Lie groups, dynamical systems, and differential geometry.
Contents: Section I: Groups and Hamiltonian structures of fluid dynamics; Section II: Topology of steady fluid flows; Section III: Topological properties of magnetic and vorticity fields; Section IV: Differential geometry of diffeomorphism groups; Section V: Kinematic fast dynamo problems; Section VI: Dynamical systems with hydrodynamical background.