Lobachevsky geometry in TTW and PW systems

Abstract

We review the classical properties of Tremblay-Turbiner-Winternitz and Post-Wintenitz systems and their relation with N-dimensional rational Calogero model with oscillator and Coulomb potentials, paying special attention to their hidden symmetries. Then we show that combining the radial coordinate and momentum in a single complex coordinate in proper way, we get an elegant description for the hidden and dynamical symmetries in these systems related with action-angle variables.

9 pages, misprint corrected, reference added. To appear in the Proceedings of the IX Symposium on Quantum Theory and Symmetries (QTS-9), July 13-18, 2015, Yerevan

References in corpus (8)

• Calogero models and nonlocal conformal transformations
• Action-angle variables for dihedral systems on the circle
• The Tremblay-Turbiner-Winternitz system on spherical and hyperbolic spaces : Superintegrability, curvature-dependent formalism and complex factorization
• Lobachevsky geometry of (super)conformal mechanics
• Runge-Lenz vector in Calogero-Coulomb problem
• Integrability and separation of variables in Calogero-Coulomb-Stark and two-center Calogero-Coulomb systems
• The structure of invariants in conformal mechanics
• Remarks on Multi-Dimensional Conformal Mechanics

Cited by in corpus (8)

• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• Symmetries in superintegrable deformations of oscillator/Coulomb systems: "holomorphic factorization"
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• New superintegrable models on spaces of constant curvature
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• More on superintegrable models on spaces of constant curvature
• Noncompact ${\mathbf{CP}}^N$ as a phase space of superintegrable systems
• Superintegrable dynamics on $H^2$ generated by coupling the Morse and Rosen-Morse potentials