World Scientific, 2004, 415 pages. This is an introductory textbook on the geometrical theory of dynamical systems, fluid flows, and certain integrable systems. The subjects are interdisciplinary and extend from mathematics, mechanics and physics to mechanical engineering, and the approach is very fundamental. The underlying concepts are based on differential geometry and theory of Lie groups in the mathematical aspect, and on transformation symmetries and gauge theory in the physical aspect. A great deal of effort has been directed toward making the description elementary, clear and concise, so that beginners will have an access to the topics. Mathematical Bases. Manifolds, Flows, Lie Groups and Lie Algebras. Geometry of Surfaces in R3. Riemannian Geometry. Dynamical Systems. Free Rotation of a Rigid Body. Water Waves and KdV Equation. Hamiltonian Systems: Chaos, Integrability and Phase Transition. Flows of Ideal Fluids. Gauge Principle and Variational Formulation. Volume-Preserving Flows of an Ideal Fluid. Motion of Vortex Filaments. Geometry of Integrable Systems. Geometric Interpretations of Sine–Gordon Equation. Integrable Surfaces: Riemannian Geometry and Group Theory.
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