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September  2009, 1(3): 317-355. doi: 10.3934/jgm.2009.1.317

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

Giuseppe Marmo 1, , Giuseppe Morandi 2, and  Narasimhaiengar Mukunda 3,

1. 

Dipartimento di Scienze Fisiche and INFN, Università di Napoli "Federico II". v. Cintia. I-80126 Naples, Italy
2. 

Dipartimento di Fisica and INFN, Università di Bologna. 6/2 v.le B.Pichat. I-40127 Bologna, Italy
3. 

Centre for Theoretical Studies, Indian Institute of Science. Bangalore 560 012, India

Received  February 2009 Revised  April 2009 Published  November 2009

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: Hamiltonian system, Hamilton-Jacobi equation, Lagrangian system, Semiclassical Approximation..
Mathematics Subject Classification: 70H20, 70S05, 70FXX, 70HX.
Citation: Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317
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