Geometric Mechanics on Riemannian Manifolds: Applications to Partial Differential Equations [Hardcover]

Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators, quartic oscillators, minimal surfaces, and Schr?dinger's, Einstein's and Newton's equations.

Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar directed at graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics. The text is enriched with good examples and exercises at the end of every chapter. It is also an ideal resource for pure and applied mathematicians and theoretical physicists working in these areas.

Introductory Chapter.- Laplace Operators on Riemannian Manifolds.- Lagrangian Formalism on Riemannian Manifolds.- Harmonic Maps from a Lagrangian Viewpoint.- Conservation Theorems.- Hamiltonian Formalism.- Hamilton-Jacobi Theory.- Minimal Hypersurfaces.- Radially Symmetric Spaces.- Fundamental Solutions for Heat Operators with Potentials.- Fundamental Solutions for Elliptic Operators.- Mechanical Curves.

...This book, which contains some very interesting ideas and results, is primarily oriented towards graduate or advanced undergraduate students in mathematics and theoretical physics with interests in differential geometry, the calculus of variations and the study of PDE's, as well as in classical and quantum mechanics. In addition, for more experienced researchers in these fields, it may be a useful resource, written in a style that makes it easily accessible to a wide audience... --- Mathematical Reviews

The differential operators which are treated in the book are among the most important, not only in the theory of partialó-