Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions

TL;DR

This paper extends the rational Calogero-Moser model in three dimensions by introducing separable, integrable, and superintegrable systems with quadratic integrals, including new potentials and reductions to lower dimensions with interesting applications.

Contribution

It introduces a broad class of integrable and superintegrable extensions of the Calogero-Moser system using separation coordinates and conformal factors, generalizing previous models.

Findings

Identification of separable systems with quadratic integrals in 3D

Extension of integrals to conformal kinetic energies

Reduction to 2D systems with Kepler and Hénon-Heiles potentials

Abstract

We consider a class of Hamiltonian systems in 3 degrees of freedom, with a particular type of quadratic integral and which includes the rational Calogero-Moser system as a particular case. For the general class, we introduce separation coordinates to find the general separable (and therefore Liouville integrable) system, with two quadratic integrals. This gives a coupling of the Calogero-Moser system with a large class of potentials, generalising the series of potentials which are separable in parabolic coordinates. Particular cases are {\em superintegrable}, including Kepler and a resonant oscillator. The initial calculations of the paper are concerned with the flat (Cartesian type) kinetic energy, but in Section \ref{sec:conflat-general}, we introduce a {\em conformal factor} $\varphi$ to $H$ and extend the two quadratic integrals to this case. All the previous results are generalised to this case. We then introduce some 2 and 3 dimensional symmetry algebras of the Kinetic energy (Killing vectors), which restrict the conformal factor. This enables us to reduce our systems from 3 to 2 degrees of freedom, giving rise to many interesting systems, including both Kepler type and H\'enon-Heiles type potentials on a Darboux-Koenigs $D_2$ background.