Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces

TL;DR

This paper extends the classical Smorodinsky-Winternitz systems to N-dimensional spheres and hyperbolic spaces, demonstrating their maximal superintegrability across different curvatures using Lie algebra methods.

Contribution

It provides a unified Lie algebra-based framework for superintegrable systems on curved spaces, preserving maximal superintegrability and deriving explicit Hamiltonians and integrals.

Findings

Maximal superintegrability is preserved on curved spaces.

Explicit Hamiltonians and integrals are derived for all curvatures.

The systems can be viewed as superpositions of oscillators with centrifugal terms.

Abstract

The classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces S^N, E^N and H^N are simultaneously approached starting from the Lie algebras so_k(N+1), which include a parametric dependence on the curvature k. General expressions for the Hamiltonian and its integrals of motion are given in terms of intrinsic geodesic coordinate systems. Each Lie algebra generator gives rise to an integral of motion, so that a set of N(N+1)/2 integrals is obtained. Furthermore, 2N-1 functionally independent ones are identified which, in turn, shows that the well known maximal superintegrability of the Smorodinsky-Winternitz system on E^N is preserved when curvature arises. On both S^N and H^N, the resulting system can be interpreted as a superposition of an "actual" oscillator and N "ideal" oscillators (for the sphere, these are alike the actual ones), which can also be understood as N "centrifugal terms"; this is the form seen in the Euclidean limiting case.