Superintegrable classical Zernike system

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 into three vertical projections of polar coordinates of sphere, polar, and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.