Factorization approach to superintegrable systems: Formalism and applications

TL;DR

This paper revisits the factorization method for superintegrable systems, applying it to classical and new anisotropic oscillators on the plane and sphere, and analyzing the Tremblay-Turbiner-Winternitz system.

Contribution

It introduces a factorization approach to construct new superintegrable systems on the sphere and revisits classical systems with this method.

Findings

Constructed new superintegrable anisotropic oscillators on the sphere.

Revisited classical anisotropic oscillator on the plane.

Analyzed the Tremblay-Turbiner-Winternitz system using factorization.

Abstract

The factorization technique for superintegrable Hamiltonian systems is revisited and applied in order to obtain additional (higher-order) constants of the motion. In particular, the factorization approach to the classical anisotropic oscillator on the Euclidean plane is reviewed, and new classical (super)integrable anisotropic oscillators on the sphere are constructed. The Tremblay-Turbiner-Winternitz system on the Euclidean plane is also studied from this viewpoint.