The Journal of Geometric Mechanics (JGM)

The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

Pages: 317 - 355, Volume 1, Issue 3, September 2009      doi:10.3934/jgm.2009.1.317

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Giuseppe Marmo - Dipartimento di Scienze Fisiche and INFN, Università di Napoli "Federico II". v. Cintia. I-80126 Naples, Italy (email)
Giuseppe Morandi - Dipartimento di Fisica and INFN, Università di Bologna. 6/2 v.le B.Pichat. I-40127 Bologna, Italy (email)
Narasimhaiengar Mukunda - Centre for Theoretical Studies, Indian Institute of Science. Bangalore 560 012, India (email)

Abstract: We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi ($HJ$) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the $HJ$ theory on the tangent bundle of a configuration manifold, the quantum $HJ$ theory, $HJ$ problems for general differential operators and the $HJ$ problem for Lie groups.

Keywords:  Hamilton-Jacobi equation, Lagrangian system, Hamiltonian system, Semiclassical Approximation.
Mathematics Subject Classification:  70H20, 70S05, 70FXX, 70HXX.

Received: February 2009;      Revised: April 2009;      Available Online: November 2009.