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Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory
This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. ...
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Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

J. Math. Phys. 46, 053510 (2005); doi:10.1063/1.1894985

Published 21 April 2005

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E. G. Kalnins
Department of Mathematics and Statistics, University of Waikato, Hamilton, New Zealand

J. M. Kress
School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia

W. Miller, Jr.
School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here we study the Stäckel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different spaces. Through the use of this tool we derive and classify for the first time all two-dimensional (2D) superintegrable systems. The underlying spaces are exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds (with zero potential) that admit at least three second order symmetries. Our derivation is very simple and quite distinct. We also show that every superintegrable system is the Stäckel transform of a superintegrable system on a constant curvature space. ©2005 American Institute of Physics
History: Received 30 November 2004; accepted 22 February 2005; published 21 April 2005
Permalink: http://link.aip.org/link/?JMAPAQ/46/053510/1
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0022-2488 (print)   1089-7658 (online)
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