dorsal/arxiv
View SchemaProgress in Classical and Quantum Variational Principles
| Authors | C. G. Gray, G. Karl, V. A. Novikov |
|---|---|
| Categories | |
| ArXiv ID | physics/0312071 |
| URL | https://arxiv.org/abs/physics/0312071 |
| DOI | 10.1088/0034-4885/67/2/R02 |
Abstract
We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schr\"{o}dinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems.
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"abstract": "We review the development and practical uses of a generalized Maupertuis\nleast action principle in classical mechanics, in which the action is varied\nunder the constraint of fixed mean energy for the trial trajectory. The\noriginal Maupertuis (Euler-Lagrange) principle constrains the energy at every\npoint along the trajectory. The generalized Maupertuis principle is equivalent\nto Hamilton\u0027s principle. Reciprocal principles are also derived for both the\ngeneralized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis\nPrinciple is the classical limit of Schr\\\"{o}dinger\u0027s variational principle of\nwave mechanics, and is also very useful to solve practical problems in both\nclassical and semiclassical mechanics, in complete analogy with the quantum\nRayleigh-Ritz method. Classical, semiclassical and quantum variational\ncalculations are carried out for a number of systems, and the results are\ncompared. Pedagogical as well as research problems are used as examples, which\ninclude nonconservative as well as relativistic systems.",
"arxiv_id": "physics/0312071",
"authors": [
"C. G. Gray",
"G. Karl",
"V. A. Novikov"
],
"categories": [
"physics.class-ph"
],
"doi": "10.1088/0034-4885/67/2/R02",
"title": "Progress in Classical and Quantum Variational Principles",
"url": "https://arxiv.org/abs/physics/0312071"
},
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