A New Superintegrable Hamiltonian

TL;DR

This paper introduces a new superintegrable Hamiltonian in three dimensions, derived from a reduction of 6D Keplerian motion, with unique integrals of motion leading to closed, periodic trajectories.

Contribution

It presents a novel superintegrable Hamiltonian generalizing Kepler's problem, with explicit integrals of motion and a new formulation in action-angle variables.

Findings

All bound trajectories are closed and strictly periodic.

The Hamiltonian admits a generalized Laplace-Runge-Lenz vector.

Integrals of motion are quartic in momenta and not from Hamilton-Jacobi separability.

Abstract

We identify a new superintegrable Hamiltonian in 3 degrees of freedom, obtained as a reduction of pure Keplerian motion in 6 dimensions. The new Hamiltonian is a generalization of the Keplerian one, and has the familiar 1/r potential with three barrier terms preventing the particle crossing the principal planes. In 3 degrees of freedom, there are 5 functionally independent integrals of motion, and all bound, classical trajectories are closed and strictly periodic. The generalisation of the Laplace-Runge-Lenz vector is identified and shown to provide functionally independent isolating integrals. They are quartic in the momenta and do not arise from separability of the Hamilton-Jacobi equation. A formulation of the system in action-angle variables is presented.