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

TL;DR

This paper introduces a unified framework for superintegrable Hamiltonians on N-dimensional constant curvature spaces, revealing common constants of motion and new superintegrable potentials through algebraic symmetry methods.

Contribution

It provides a universal approach to superintegrability on curved spaces, identifying common constants of motion and new potentials via sl(2) Poisson coalgebra symmetry.

Findings

Unified set of constants of motion for superintegrable systems

Explicit identification of integrals in involution including the Hamiltonian

Introduction of new classes of superintegrable potentials

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 here presented are a consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians, together with an appropriate use of the phase spaces associated to Poincare and Beltrami coordinates.