On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

Abstract

We extend the investigation of three-dimensional Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry, namely the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. More precisely, we focus on linear and some special cases of quadratic superintegrability. In the linear case, no new superintegrable system arises. In the quadratic case, we found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases and another one at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems. These two systems are linked respectively with the caged and harmonic oscillators without magnetic fields through a time-dependent canonical transformation.

Superintegrability of separable systems with magnetic field: the cylindrical case

“…As such systems with additional first order integrals were classified in [21] (and only systems already found in [8,12] were shown to exist under this assumption), we have assumed that the additional independent integral is quadratic. We have also excluded systems contained in [13,12] and their special cases. With these restrictions, only systems obtained through Propositions 4.1 have been found in Cases I and II, the ones in Case I separating in Cartesian coordinates and therefore known from [12].…”

Linearity of minimally superintegrable systems in a static electromagnetic field

“…This system represents the intersection between the circular parabolic case and the spherical case with a magnetic field. In [6], Bertrand et al continued the study of three-dimensional integrable systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry. Two new minimally superintegrable systems admitting an additional quadratic integral were presented and they represent the intersection of more than one integrable case.…”

“…Two new minimally superintegrable systems admitting an additional quadratic integral were presented and they represent the intersection of more than one integrable case. We do not consider the superintegrable systems admitting an additional linear integral as determined in [5,6] since they are subcases of the systems we have already investigated in our present paper.…”

“…In [28], a systematic study of integrable and superintegrable systems in the presence of a magnetic field in three-dimensional Euclidean space was initiated, and then continued in several papers [5,6,26,27]. All of those systems are autonomous, integrable and separable in at least one set of coordinates.…”

“…(In [5,6,26,27], the mass and the charge of the particle have been rescaled to 1 and −1, respectively. We will follow this convention throughout this paper.)…”