Hamilton-Jacobi theory and the relativistic action variable

H. Rose

Fritz Rohrlich

Oliver Davis Johns

Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.

Julian Schwinger

Robert A. Leacock, Michael J. Padgett

A quantum action variable is defined in the context of a quantum Hamilton-Jacobi theory. The action variable can be used to find the exact bound-state energy levels of a quantum system without solving an equation of motion for the system wave functions.

Richard F. Arens