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Analytical Mechanics for Relativity and Quantum Mechanics$
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Oliver Johns

Print publication date: 2011

Print ISBN-13: 9780191001628

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780191001628.001.0001

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Hamilton-Jacobi Therory

Hamilton-Jacobi Therory

(p.477) 21 Hamilton-Jacobi Therory
Analytical Mechanics for Relativity and Quantum Mechanics

Oliver Davis Johns

Oxford University Press

This chapter focuses on the Hamilton-Jacobi theory, the apotheosis of Lagrangian and Hamiltonian mechanics. Action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria, and are the solutions of a non-linear, first-order partial differential equation called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. An entire 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 connection is discussed in detail in the chapter, together with the Bohm hidden variable model and Feynman path integral method that are derived from it.

Keywords:   action functions, Hamilton-Jacobi theory, mechanical system, Hamilton-Jacobi equation, Schroedinger formulation, quantum mechanics

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