New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases

TL;DR

This paper explores new classes of integrable and superintegrable magnetic systems in three-dimensional Euclidean space, identifying three new cylindrical cases, one superintegrable system, and analyzing their mathematical properties and physical relevance.

Contribution

It introduces three new integrable magnetic systems extending cylindrical symmetry and investigates superintegrability, revealing a unique non-separable superintegrable system.

Findings

Found three new integrable systems in the generalized cylindrical case.

Identified no new integrable systems in the spherical case.

Discovered a minimally superintegrable system modeling a helical undulator.

Abstract

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton-Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.