• Corpus ID: 251468185

On the generalization of classical Zernike system

  title={On the generalization of classical Zernike system},
  author={Cezary Gonera and Joanna Gonera and Piotr Kosiński},
We generalize the results obtained recently (Nonlinearity 36 (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H = ~p 2 + F ( ~q · ~p ), ~q, ~p ∈ 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 ~q, ~p ∈ R n is sketched. 



Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and N∈N . For N = 2,

From Free Motion on a 3-Sphere to the Zernike System of Wavefronts Inside a Circular Pupil

  • K. Wolf
  • Physics
    Journal of Physics: Conference Series
  • 2020
Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into

Spherical geometry, Zernike’s separability, and interbasis expansion coefficients

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include

Classical and Quantum Super-Integrability: From Lissajous Figures to Exact Solvability

  • A. Fordy
  • Mathematics
    Physics of Atomic Nuclei
  • 2018
The first part of this paper explains what super-integrability is and how it differs in the classical and quantum cases. This is illustrated with an elementary example of the resonant harmonic

Interbasis expansions in the Zernike system

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical

New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set

Quantum superintegrable Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike

Superintegrable classical Zernike system

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

Spherical Geometry

  • J. Ratcliffe
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 1883
The object of this paper is to bring together the principal properties of figures described on the surface of the sphere that can be established without the use of Solid Geometry or of Trigonometry.

Diffraction theory of the cut procedure and its improved form

  • the phase contrast method, Physica 1
  • 1934