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Journal of Mathematical Physics / Volume 53 / Issue 1 / ARTICLES / Classical Mechanics and Classical Fields

J. Math. Phys. 53, 012901 (2012); http://dx.doi.org/10.1063/1.3676075 (12 pages)

Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

Ian Marquette

School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia

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(Received 2 October 2011; accepted 14 December 2011; published online 6 January 2012)

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.

© 2012 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. QUANTUM AND CLASSICAL SYSTEMS WITH LADDER OPERATORS
    1. Classification of systems in classical and quantum mechanics with ladder operators
  3. CLASSICAL SYSTEMS WITH LADDER OPERATORS OF ORDER 3
  4. SYSTEMS WITH FOURTH-ORDER LADDER OPERATORS
  5. NEW SUPERINTEGRABLE SYSTEMS IN CLASSICAL MECHANICS
    1. Two new families of superintegrable systems
  6. CONCLUSION

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KEYWORDS and PACS

Keywords

harmonic oscillators, integral equations, Poisson equation, polynomial matrices

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3676075

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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