Superintegrability on N-dimensional spaces of constant curvature from so(N+1) and its contractions

TL;DR

This paper constructs and analyzes superintegrable Hamiltonian systems on N-dimensional constant curvature spaces using the algebra so(N+1), identifying new maximally superintegrable models including generalizations of Smorodinsky-Winternitz and Kepler-Coulomb systems.

Contribution

It introduces a unified algebraic framework to generate superintegrable systems on various N-dimensional spaces and explicitly constructs new maximally superintegrable models.

Findings

Identified a family of quasi-maximally superintegrable systems with 2N-3 constants of motion.

Derived two maximally superintegrable Hamiltonians generalizing known systems.

Explicitly expressed all systems and integrals in a unified coordinate framework.

Abstract

The Lie-Poisson algebra so(N+1) and some of its contractions are used to construct a family of superintegrable Hamiltonians on the ND spherical, Euclidean, hyperbolic, Minkowskian and (anti-)de Sitter spaces. We firstly present a Hamiltonian which is a superposition of an arbitrary central potential with N arbitrary centrifugal terms. Such a system is quasi-maximally superintegrable since this is endowed with 2N-3 functionally independent constants of the motion (plus the Hamiltonian). Secondly, we identify two maximally superintegrable Hamiltonians by choosing a specific central potential and finding at the same time the remaining integral. The former is the generalization of the Smorodinsky-Winternitz system to the above six spaces, while the latter is a generalization of the Kepler-Coulomb potential, for which the Laplace-Runge-Lenz N-vector is also given. All the systems and constants of the motion are explicitly expressed in a unified form in terms of ambient and polar coordinates as they are parametrized by two contraction parameters (curvature and signature of the metric).