Ternary Poisson algebra for the non degenerate three dimensional Kepler Coulomb potential

TL;DR

This paper explores the algebraic structure of integrals of motion in superintegrable three-dimensional Kepler-Coulomb systems, revealing a ternary Poisson algebra framework that generalizes known quadratic symmetries.

Contribution

It introduces a ternary Poisson algebra structure for the Kepler-Coulomb potential, extending the algebraic understanding of superintegrability in three dimensions.

Findings

The Poisson algebra of integrals is quadratic and ternary in form.

The Kepler-Coulomb potential exhibits a sixth integral of motion.

The algebraic structure aligns with non-degenerate superintegrable systems.

Abstract

In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. The Kepler Coulomb potential that was introduced by Verrier and Evans is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the non degenerate case of systems with quadratic integrals.