Adding Potentials to Superintegrable Systems with Symmetry

TL;DR

This paper explores how to incorporate potential functions into superintegrable Hamiltonian systems with symmetry, revealing connections to classical problems like Kepler and isotropic oscillator.

Contribution

It extends previous work by adding potentials to symmetric Hamiltonians, linking superintegrability with classical systems and identifying new separable potentials.

Findings

Separable potentials reduce to 3 or 4 parameter families.

Connections established between Darboux-Koenigs and classical Hamiltonians.

Potential functions compatible with symmetry are characterized.

Abstract

In previous work, we have considered Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. Previously our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the 3 dimensional space reduce to 3 or 4 parameter potentials for Darboux-Koenigs Hamiltonians. Other 3D coordinate systems reveal connections between Darboux-Koenigs and other well known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.