Superintegrable systems on spaces of constant curvature

Abstract

Construction and classification of 2D superintegrable systems (i.e. systems admitting, in addition to two global integrals of motion guaranteeing the Liouville integrability, the third global and independent one) defined on 2D spaces of constant curvature and separable in the so called geodesic polar coordinates are presented. The method proposed is applicable to any value of curvature including the case of Euclidean plane, the 2-sphere and the hyperbolic plane. The mathematic used is essentially "physical", in particular it refers to the very elegant technique of action-angle variables and perturbation theory so most of our mathematical formulas have a clear physical meaning. The main result can be considered as a kind of generalization of the Bertrand's theorem on 2D spaces of constant curvature and it covers most of known separable and superintegrable models on such spaces ( in particular, the so-called Tremblay-Turbiner-Winternitz (TTW) and Post-Winternitz (PW) models which have recently attracted some in-

Superintegrable deformations of superintegrable systems: Quadratic superintegrability and higher-order superintegrability

“…Fris et al studied [1] the two-dimensional Euclidean systems, which admit separability in more than one coordinate systems and obtained four families of potentials V r , r = a, b, c, d, possessing three functionally independent integrals of motion (they were mainly interested in the quantum two-dimensional Schrödinger equation but the results obtained are also valid at the classical level). Then other authors studied similar problems on higher-dimensional Euclidean spaces [2]- [4], on two-dimensional spaces with a pseuo-Euclidean metric (Drach potentials) [5]- [8], or on curved spaces [9]- [15] (see [16] for a recent review on superintegrability that includes a long list of references).…”

The Post–Winternitz system on spherical and hyperbolic spaces: A proof of the superintegrability making use of complex functions and a curvature-dependent formalism

“…first studied by Tremblay, Turbiner, and Winternitz [2,3], and then by other authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It is only separable in polar coordinates and therefore the additional third integral, that is of higher order in the momenta, cannot be obtained by making use of techniques related with separability.…”

“…The first proof of the superintegrability of this new potential was given in [18] by relating V pw with V ttw via a coupling constant metamorphosis transformation [19]. These two systems, TTW and PW systems, have been recently studied not only in the Euclidean plane but also on spaces of constant curvature κ [7,12,15,17]. It is known that the existence of superintegrability of a system can be studied making use of different approaches (Hamilton-Jacobi formalism and action-angle variables, proof that all bounded classical trajectories are closed, exact solvability, degenerate quantum energy levels).…”

New superintegrable models on spaces of constant curvature

“…For a convenience of the reader as well as to fix the notation we start with brief recapitulation of the main ideas and results of Ref. [32]. Then, we explain how the implicit and transcendental, in general, equation on angular potentials can be solved in terms of elementary functions.…”