Hamilton–Jacobi Theory

Abstract

Besides the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics, there is yet a fourth approach, known as Hamilton–Jacobi theory, which is part of Hamiltonian mechanics. This approach is the subject of the present chapter. In Hamilton–Jacobi theory, the central equation capturing the dynamics of the mechanical system is the Hamilton–Jacobi equation which is a first-order, non-linear partial differential equation. Remarkably and contrary to the other formulations of classical mechanics, the entire multi-dimensional dynamics is described by a single equation. Even for relatively simple mechanical systems, the corresponding Hamilton–Jacobi equation can be hard or even impossible to solve analytically. However, its virtue lies in the fact that it offers a useful, alternative way of identifying conserved quantities even in cases where the Hamilton–Jacobi equation itself cannot be solved directly. In addition, Hamilton–Jacobi theory has played an important historical role in the development of quantum mechanics, since the Hamilton–Jacobi equation can be viewed as a precursor to the Schrödinger equation (Bates, Weinstein, Berkeley Math. Lect. Notes 8:1–137 1997, [3]), (Goldstein, Classical Mechanics, Addison-Wesley, Menlo Park, 1980, [7]), (Sakurai, Modern Quantum Mechanics, Benjamin Cummings Publishing Company, Reading, 1985, [18]).