Maximal superintegrability on N-dimensional curved spaces

TL;DR

This paper presents a unified algebraic framework for constructing maximally superintegrable systems on N-dimensional curved spaces, including spheres, Euclidean, and hyperbolic geometries, demonstrating their superintegrability across different curvatures.

Contribution

It introduces explicit algebraic expressions for Hamiltonians and integrals of motion on ND curved spaces, generalizing the Smorodinsky-Winternitz systems and proving their maximal superintegrability.

Findings

Explicit formulas for Hamiltonians and integrals of motion on ND spaces.

Superintegrability holds for all curvatures in Euclidean spaces.

Potential interpreted as a superposition of oscillators on the sphere.

Abstract

A unified algebraic construction of the classical Smorodinsky-Winternitz systems on the ND sphere, Euclidean and hyperbolic spaces through the Lie groups SO(N+1), ISO(N), and SO(N,1) is presented. Firstly, general expressions for the Hamiltonian and its integrals of motion are given in a linear ambient space $R^{N+1}$, and secondly they are expressed in terms of two geodesic coordinate systems on the ND spaces themselves, with an explicit dependence on the curvature as a parameter. On the sphere, the potential is interpreted as a superposition of N+1 oscillators. Furthermore each Lie algebra generator provides an integral of motion and a set of 2N-1 functionally independent ones are explicitly given. In this way the maximal superintegrability of the ND Euclidean Smorodinsky-Winternitz system is shown for any value of the curvature.