Superintegrable deformations of superintegrable systems : Quadratic superintegrability and higher-order superintegrability

TL;DR

This paper explores how known superintegrable Hamiltonian systems can be deformed through a parameter to produce new systems with continuous constants of motion, examining quadratic and higher-order superintegrability, and relating findings to TTW and PW systems.

Contribution

It introduces a family of deformed Hamiltonians with continuous parameters that preserve superintegrability, extending the understanding of quadratic and higher-order superintegrability.

Findings

Deformation preserves superintegrability with continuous parameters.

Established connections between higher-order superintegrability and TTW/PW systems.

Identified new constants of motion for deformed Hamiltonians.

Abstract

The superintegrability of four Hamiltonians $\tilde{H_r} = \lambda\, H_r$, $r=a,b,c,d$, where $H_r$ are known Hamiltonians and $\lambda$ is a certain function defined on the configuration space and depending of a parameter $\kappa$, is studied. The new Hamiltonians, and the associated constants of motion $J_{ri}$, $i=1,2,3$, are continous functions of the parameter $\kappa$. The first part is concerned with separability and quadratic superintegrability (the integrals of motion are quadratic in the momenta) and the second part is devoted to the existence of higher-order superintegrability. The results obtained in the second part are related with the TTW and the PW systems.