The Hamilton–Jacobi Theory

Abstract

We are able to solve Hamilton’s equations only in a few cases. Consequently, we must search for information about their solutions following other paths. In this chapter, we show that the Hamilton–Jacobi theory allows, at least in principle, to determine a set of canonical coordinates in which the Hamiltonian equations can be integrated at once. However, these coordinates can be determined solving the Hamilton–Jacobi nonlinear partial differential equation. Usually, to solve this equation is more difficult than to integrate Hamiltonian equations, but Jacobi proposed the method of separated variables that in some cases allows to solve the Hamilton–Jacobi equation.