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 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.

Submitted to J Math Phys

References in corpus (2)

• Two important examples of nonlinear oscillators
• Separation of variables and Lie algebra contractions. Applications to special functions

Cited by in corpus (7)

• Quantum superintegrable Zernike system
• New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion
• Interbasis expansions in the Zernike system
• Zernike functions, rigged Hilbert spaces and potential applications
• Classical and Quantum Super-integrability: From Lissajous Figures to Exact Solvability
• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum