Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature

Abstract

An infinite family of classical superintegrable Hamiltonians defined on the N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a common set of (2N − 3) functionally independent constants of the motion. Among them, two different subsets of N integrals in involution (including the Hamiltonian) can always be explicitly identified. As particular cases, we recover in a straightforward way most of the superintegrability properties of the Smorodinsky–Winternitz and generalized Kepler–Coulomb systems on spaces of constant curvature and we introduce as well new classes of (quasi-maximally) superintegrable potentials on these spaces. Results presented here are a consequence of the Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated with Poincaré and Beltrami coordinates.