A non-linear Oscillator with quasi-Harmonic behaviour: two- and $n$-dimensional Oscillators

TL;DR

This paper investigates a two-dimensional nonlinear oscillator, demonstrating its super-integrability, explicit solutions, and generalization to n dimensions, with connections to harmonic oscillators on curved spaces.

Contribution

It introduces a super-integrable nonlinear oscillator model in two and n dimensions, extending previous one-dimensional studies and linking it to harmonic oscillators on curved geometries.

Findings

All bounded motions are quasiperiodic oscillations.

Unbounded motions are hyperbolic functions.

The system is related to harmonic oscillators on curved spaces.

Abstract

A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a two-dimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of $n$ degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere $S^2$ and hyperbolic plane $H^2$, is discussed.