A new proof of the higher-order superintegrability of a noncentral oscillator with inversely quadratic nonlinearities

TL;DR

This paper presents a new proof demonstrating the higher-order superintegrability of a noncentral harmonic oscillator with inverse quadratic nonlinearities, extending the understanding of integrable systems with centrifugal terms.

Contribution

It provides a novel proof of superintegrability for a class of noncentral oscillators and generalizes the results to systems with multiple degrees of freedom.

Findings

Explicit constants of motion for the nonlinear system are derived.

Higher-order superintegrability is established through a deformation of the quadratic complex equation.

Conditions for superintegrability in systems with multiple degrees of freedom are identified.

Abstract

The superintegrability of a rational harmonic oscillator (non-central harmonic oscillator with rational ratio of frequencies) with non-linear "centrifugal" terms is studied. In the first part, the system is directly studied in the Euclidean plane; the existence of higher-order superintegrability (integrals of motion of higher order than 2 in the momenta) is proved by introducing a deformation in the quadratic complex equation of the linear system. The constants of motion of the nonlinear system are explicitly obtained. In the second part, the inverse problem is analyzed in the general case of $n$ degrees of freedom; starting with a general Hamiltonian $H$, and introducing appropriate conditions for obtaining superintegrability, the particular "centrifugal" nonlinearities are obtained.