Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

TL;DR

This paper constructs a family of superintegrable Hamiltonian systems on various 3D spaces of constant curvature and signature, identifying special cases with maximal superintegrability and deriving associated conserved quantities like the Laplace-Runge-Lenz vector.

Contribution

It introduces a unified framework for superintegrable systems on 3D curved and relativistic spaces, including new potentials generalizing known models and deriving their conserved quantities.

Findings

Unified description of superintegrable systems across different spaces

Identification of maximally superintegrable Hamiltonians

Explicit expressions for conserved quantities like Laplace-Runge-Lenz vector

Abstract

A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.