Maximal superintegrability of the generalized Kepler--Coulomb system on N-dimensional curved spaces

TL;DR

This paper extends the superintegrability of the Kepler-Coulomb system with centrifugal terms from 3D Euclidean space to N-dimensional curved spaces, revealing a rich structure of integrals of motion.

Contribution

It generalizes the superintegrability of the Kepler-Coulomb system to N-dimensional curved spaces using a unified symmetry approach, identifying new integrals of motion.

Findings

The Hamiltonian has (2N-1) independent integrals of motion.

One integral is quartic, others are quadratic.

The transition from Euclidean to curved spaces is fully characterized.

Abstract

The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the momenta. In this paper we present the generalization of this result to the ND spherical, hyperbolic and Euclidean spaces by making use of a unified symmetry approach that makes use of the curvature parameter. The resulting Hamiltonian, formed by the (curved) Kepler-Coulomb potential together with N centrifugal terms, is shown to be endowed with (2N-1) functionally independent integrals of the motion: one of them is quartic and the remaining ones are quadratic. The transition from the proper Kepler-Coulomb potential, with its associated quadratic Laplace-Runge-Lenz N-vector, to the generalized system is fully described. The role of spherical, nonlinear (cubic), and coalgebra symmetries in all these systems is highlighted.