Variational Methods

Abstract

This chapter is a modification and extension, of some of the material in [15, Chap. 9], to our specific needs. The emphasis will be on the Lagrange and Hamilton canonical equations of motion, with applications to wave propagation in electromagnetic media. For the convenience of the reader, some of the essential features of the calculus of variations, as well as D’Alembert’s principle, Hamilton’s principle and other variational principles, will be reviewed in the context of phase space. This is the setting for a proper understanding of the Hamilton—Jacobi theory which gives us a deep insight into the partial differential equations (PDEs) of wave propagation. For a more thorough treatment, the reader is referred to the above reference, and to the standard works on the calculus of variations such as [5], for a more refined mathematical treatment of this subject which involves existence theorems, and so forth.