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Superintegrability in a two-dimensional space of nonconstant curvature

J. Math. Phys. 43, 970 (2002); doi:10.1063/1.1429322

Issue Date: February 2002

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E. G. Kalnins and J. M. Kress
Department of Mathematics, University of Waikato, Private Bag 3105, Hamilton, New Zealand

P. Winternitz
Centre de recherches mathématiques, Université de Montréal, C.P. 6128-CV, Montréal, Québec H3C 3J7, Canada
A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined. ©2002 American Institute of Physics.
History: Received 21 August 2001; accepted 1 November 2001
Permalink: http://link.aip.org/link/?JMAPAQ/43/970/1
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KEYWORDS and PACS

Keywords
PACS
  • 45.05.+x
    Classical mechanics of discrete systems General theory of classical mechanics of discrete systems
  • 02.40.-k
    Mathematical methods in physics Geometry, differential geometry, and topology
  • YEAR: 2002

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