On the generalization of classical Zernike system

TL;DR

This paper proves the superintegrability of a broad class of Hamiltonian systems involving a function of position and momentum dot product, providing explicit integrals of motion and extending results to higher dimensions.

Contribution

It offers a simple proof of superintegrability for Hamiltonians with an arbitrary analytic function of the dot product of position and momentum, and sketches higher-dimensional generalizations.

Findings

Explicit integral of motion constructed

Superintegrability proven for any analytic function F

Extension to higher dimensions sketched

Abstract

We generalize the results obtained recently (Nonlinearity \underline{36} (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian $H=\vec{p}\,^{2}+F(\vec{q}\cdot\vec{p})$, $\vec{q}, \vec{p}\in\mathbb{R}^{2}$, for any analytic function $F$. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial $F$. The generalization to the case $\vec{q}, \vec{p}\in \mathbb{R}^{n}$ is sketched.