CHAPTER II - HAMILTON-JACOBI THEORY

Abstract

This chapter provides an overview of the Hamilton–Jacobi theory. To the given manifold M, there corresponds a global coordinate-independent Hamilton function H, in a similar way as to M, there corresponds a global coordinate-independent Lagrange function L. The vectors y are the representatives of covariant vectors. Any admissible transformation of coordinates transforms Lagrange's equations of motion into equations of the same form. The one-to-one correspondence between Lagrange's and Hamilton's equations suggests that such a transformation transforms Hamiltonian equations into Hamiltonian equations. The canonical transformations of local coordinates defined in appropriate neighborhoods of a given point (x0,y0) form a group. The set of the time-dependent canonical transformations is a subgroup of this group. The local canonical transformations of multiplier l form a normal subgroup of the group of all local canonical transformations. Canonical transformations of multiplier l are called completely canonical.