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Solutions of the Hamilton-Jacobi Equation and Their Connection to Quantum Mechanics

A. Pareek

Published Jan 1, 1988• 0 citations
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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.

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Abstract 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.

•May 19, 2011•0 citations
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.
Abstract 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.