Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 008, 24 pages      arXiv:0901.3081      https://doi.org/10.3842/SIGMA.2009.008

Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

Ernest G. Kalnins a, Jonathan M. Kress b, Willard Miller Jr. c and Sarah Post c
a) Department of Mathematics, University of Waikato, Hamilton, New Zealand
b) School of Mathematics, The University of New South Wales, Sydney NSW 2052, Australia
c) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA

Received November 26, 2008, in final form January 14, 2009; Published online January 20, 2009

Abstract
The structure theory for the quadratic algebra generated by first and second order constants of the motion for 2D second order superintegrable systems with nondegenerate (3-parameter) and or 2-parameter potentials is well understood, but the results for the strictly 1-parameter case have been incomplete. Here we work out this structure theory and prove that the quadratic algebra generated by first and second order constants of the motion for systems with 4 second order constants of the motion must close at order three with the functional relationship between the 4 generators of order four. We also show that every 1-parameter superintegrable system is Stäckel equivalent to a system on a constant curvature space.

Key words: superintegrability; quadratic algebras.

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