Superintegrable Deformations of the Smorodinsky-Winternitz Hamiltonian

TL;DR

This paper introduces a method to create superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian using quantum deformations of Poisson algebra symmetries, revealing connections between coalgebra symmetry and superintegrability.

Contribution

It presents a constructive procedure for superintegrable Hamiltonian deformations based on quantum algebra techniques, expanding the understanding of symmetry in integrable systems.

Findings

Established a link between coalgebra symmetry and superintegrability.

Developed new integrable deformations of Smorodinsky-Winternitz systems.

Showed applicability of comodule algebra symmetry in constructing deformations.

Abstract

A constructive procedure to obtain superintegrable deformations of the classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this example, the general connection between coalgebra symmetry and quasi-maximal superintegrability is analysed. The notion of comodule algebra symmetry is also shown to be applicable in order to construct new integrable deformations of certain Smorodinsky-Winternitz systems.