Superintegrable 2D systems in magnetic fields with a parabolic type integral

TL;DR

This paper investigates two-dimensional superintegrable systems in magnetic fields, focusing on cases where integrals of motion are quadratic in momenta, and confirms that only systems with constant magnetic and electrostatic fields are superintegrable under these conditions.

Contribution

It extends previous work by analyzing systems with parabolic type integrals, confirming the uniqueness of superintegrable systems with constant fields in the Euclidean plane.

Findings

Only superintegrable systems with constant magnetic and electrostatic fields exist under quadratic integral assumptions.

The study confirms previous classifications for Cartesian and polar integrals in the presence of magnetic fields.

New analysis of systems with parabolic type integrals supports the uniqueness result.

Abstract

We consider the problem on the existence of two dimensional superintegrable systems in the presence of a magnetic field in the two dimensional Euclidean space. We assume the existence of two integrals of motion, besides the Hamiltonian, that are quadratic polynomials in the momenta. This problem was already studied in the cases where one integral is of Cartesian or polar type [J. B\'erub\'e, and P. Winternitz, J. Math. Phys., 45(5): 1959-1973, 2004]. We continue the investigation by assuming that one of the integrals is of parabolic type and the second integral is of elliptic or (''non-standard'') parabolic type, confirming so far that, on the Euclidean plane, the only two dimensional superintegrable system with quadratic integrals is the one with constant magnetic field and constant electrostatic potential.