Global versus local superintegrability of nonlinear oscillators

TL;DR

This paper explores the relationship between Lie point symmetries and superintegrability in Hamiltonian systems, demonstrating that standard symmetry methods can be extended to find all first integrals even in complex nonlinear oscillators.

Contribution

It shows that Lie point symmetries are not directly related to superintegrability but can be extended to obtain all first integrals in nonlinear oscillators with non-constant curvature.

Findings

Superintegrability is not directly linked to the algebra of variational symmetries.

Standard point symmetry methods can be extended to find all first integrals.

The approach is applied to a nonlinear oscillator on a curved space with spherical symmetry.

Abstract

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.