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Analytical Mechanics for Relativity and Quantum Mechanics$
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Oliver Johns

Print publication date: 2005

Print ISBN-13: 9780198567264

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198567264.001.0001

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Hamilton–Jacobi Theory

Hamilton–Jacobi Theory

Chapter:
(p.461) 19 Hamilton–Jacobi Theory
Source:
Analytical Mechanics for Relativity and Quantum Mechanics
Author(s):

Oliver Johns

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198567264.003.0019

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.

Keywords:   Hamilton-Jacobi theory, canonical transformations, action functions, Schroedinger formulation, quantum mechanics, classical mechanics, complete integrals, quantum Cauchy problem

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