Description
Book SynopsisThis is an introductory textbook on the geometrical theory of dynamical systems, fluid flows and certain integrable systems. The topics are interdisciplinary and extend from mathematics, mechanics and physics to mechanical engineering, and the approach is very fundamental. The main theme of this book is a unified formulation to understand dynamical evolutions of physical systems within mathematical ideas of Riemannian geometry and Lie groups by using well-known examples. Underlying mathematical concepts include transformation invariance, covariant derivative, geodesic equation and curvature tensors on the basis of differential geometry, theory of Lie groups and integrability. These mathematical theories are applied to physical systems such as free rotation of a top, surface wave of shallow water, action principle in mechanics, diffeomorphic flow of fluids, vortex motions and some integrable systems.In the latest edition, a new formulation of fluid flows is also presented in a unified fashion on the basis of the gauge principle of theoretical physics and principle of least action along with new type of Lagrangians. A great deal of effort has been directed toward making the description elementary, clear and concise, to provide beginners easy access to the topics.
Table of ContentsMathematical 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 of Fluid Flows; 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.
