Superintegrability in a twodimensional space of nonconstant curvature
Abstract
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 twodimensional 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 HamiltonJacobi and Schrödinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 February 2002
 DOI:
 10.1063/1.1429322
 arXiv:
 arXiv:mathph/0108015
 Bibcode:
 2002JMP....43..970K
 Keywords:

 45.05.+x;
 02.40.k;
 General theory of classical mechanics of discrete systems;
 Geometry differential geometry and topology;
 Mathematical Physics;
 Exactly Solvable and Integrable Systems;
 37K05 70H20
 EPrint:
 27 pages