# Superintegrable classical Zernike system

## TL;DR

This paper analyzes the classical Zernike system, revealing its superintegrability through higher-order invariants and separation of variables in various coordinate systems, linking wavefront aberration classification to advanced Hamiltonian dynamics.

## Contribution

It demonstrates that the classical Zernike system is superintegrable, with explicit invariants and separability properties, connecting wavefront aberration modeling to integrable Hamiltonian systems.

## Key findings

- Trajectories are closed ellipses due to higher-order invariants.
- The system's Hamilton-Jacobi action separates in multiple coordinate systems.
- The Zernike system belongs to the class of superintegrable systems.

## Abstract

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates in three vertical projections of polar coordinates of a sphere, polar and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.