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 interest).

27 pages, no figure,; one reference added

References in corpus (3)

• Classical and Quantum Superintegrability with Applications
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• On two-dimensional integrable models with a cubic or quartic integral of motion

Cited by in corpus (16)

• Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces : Superintegrability, curvature-dependent formalism and complex factorization
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability
• New superintegrable models on spaces of constant curvature
• Transformation of the Stackel matrices preserving superintegrability
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Curvature as an integrable deformation
• More on superintegrable models on spaces of constant curvature
• Superintegrability of 3-dimensional Hamiltonian systems with conformally Euclidean metrics. Oscillator-related and Kepler-related systems
• Superintegrable systems with position dependent mass: master symmetry and action-angle methods
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$
• Superintegrable dynamics on $H^2$ generated by coupling the Morse and Rosen-Morse potentials
• Superintegrability of the Post-Winternitz system on spherical and hyperbolic spaces