\"本书讲述了：Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of math-metical science. Many important achievements in this field are based on profound theories rather than on experiments. In ram, those hydro dynamical theories stimulated developments in the domains of pure mathematics, such as complex analysis, topology, stability theory, bifurcation theory, and completely integral dynamical systems. In spite of all this acknowledged success, hydrodynamics with its spec-tabular empirical laws remains a challenge for mathematicians.\"

\"Preface Acknowledgments I.Group and Hamiltonian Structures of Fluid Dynamics 1.Symmetry groups for a rigid body and an ideal fluid 2.Lie groups, Lie algebras, and adjoint representation 3.Coadjoint representation of a Lie group 3.A.Definition of the coadjoint representation 3.B.Dual of the space of plane divergence-free vector fields 3.C.The Lie algebra of divergence-free vector fields and its dual in arbitrary dimension 4.Left-invariant metrics and a rigid body for an arbitrary group 5.Applications to hydrodynamics 6.Hamiltonian structure for the Euler equations 7.Ideal hydrodynamics on Riemannian manifolds 7.A.The Euler hydrodynamic equation on manifolds 7.B.Dual space to the Lie algebra of divergence-free fields 7.C.Inertia operator of an n-dimensional fluid 8.Proofs of theorems about the Lie algebra of divergence-free fields and its dual 9.Conservation laws in higher-dimensional hydrodynamics 10.The group setting of ideal magnetohydrodyuamics 10.A.Equations of magnetohydrodynamics and the Kirchhoff equations 10.B.Magnetic extension of any Lie group 10.C.Hamiltonian formulation of the Kirchhoff and magnetohydrodynamics equations 11.Finite-dimensional approximations of the Euler equation 11.A.Approximations by vortex systems in the plane 11.B.Nonintegrability of four or more point vortices 11.C.Hamiltonian vortex approximations in threedimensions 11.D.Finite-dimensional approximations of diffeomorphismgroups 12.The Navier-Stokes equation from the group viewpoint II.Topology of Steady Fluid Flows 1.Classification of three-dimensional steady flows 1.A.Stationary Euler solutions and Bernoulli functions 1.B.Structural theorems 2.Variational principles for steady solutions and applications to two-dimensional flows 2.A.Minimization of the energy 2.B.The Dirichlet problem and steady flows 2.C.Relation of two variational principles 2.D.Semigroup variational principle fo\"