Superintegrability in classical mechanics
Abstract
Superintegrable Hamiltonians in three degrees of freedom possess more than three functionally independent globally defined and singlevalued integrals of motion. Some familiar examples, such as the Kepler problem and the harmonic oscillator, have been known since the time of Laplace. Here, a classification theorem is given for superintegrable potentials with invariants that are quadratic polynomials in the canonical momenta. Such systems must possess separable solutions to the HamiltonJacobi equation in more than one coordinate system. There are 11 coordinate systems for which the HamiltonJacobi equation separates in openR^{3}. One coordinate system may be arbitrarily rotated or translated with respect to the other, yielding 66 distinct cases. In each case, the differential equations for separability in the two coordinates are integrated to give a complete list of all superintegrable potentials with four or five quadratic integrals. The tableswhich may be consulted independently of the main body of the paperlist the distinct superintegrable potentials, the separating coordinates, and the isolating integrals of the motion. If there exist five isolating integrals, then all finite classical trajectories are closed; if only four, then the trajectories are restricted to twodimensional surface. An extraordinary consequence of the work is the discovery of perturbations to both the Kepler problem and the harmonic oscillator that do not destroy the fragile degeneracy. The perturbed systems still have five isolating integrals of the motion.
 Publication:

Physical Review A
 Pub Date:
 May 1990
 DOI:
 10.1103/PhysRevA.41.5666
 Bibcode:
 1990PhRvA..41.5666E
 Keywords:

 03.20.+i