On two superintegrable nonlinear oscillators in N dimensions

TL;DR

This paper studies a class of superintegrable nonlinear oscillators in N dimensions, revealing their geometric structure, constants of motion, and connections to curved spaces and quantum models.

Contribution

It demonstrates the Stackel equivalence of a superintegrable Hamiltonian to free Euclidean motion and identifies three underlying manifolds, introducing new nonlinear oscillator models.

Findings

Identified three different nonlinear oscillator models from a superintegrable Hamiltonian.

Established the Stackel equivalence to free Euclidean motion, aiding in finding constants of motion.

Discussed quantization and connections to spherical confinement models.

Abstract

We consider the classical superintegrable Hamiltonian system given by $H=T+U={p^2}/{2(1+\lambda q^2)}+{{\omega}^2 q^2}/{2(1+\lambda q^2)}$, where U is known to be the "intrinsic" oscillator potential on the Darboux spaces of nonconstant curvature determined by the kinetic energy term T and parametrized by {\lambda}. We show that H is Stackel equivalent to the free Euclidean motion, a fact that directly provides a curved Fradkin tensor of constants of motion for H. Furthermore, we analyze in terms of {\lambda} the three different underlying manifolds whose geodesic motion is provided by T. As a consequence, we find that H comprises three different nonlinear physical models that, by constructing their radial effective potentials, are shown to be two different nonlinear oscillators and an infinite barrier potential. The quantization of these two oscillators and its connection with spherical confinement models is briefly discussed.