Superintegrable 3D systems in a magnetic field corresponding to Cartesian separation of variables

Abstract

We consider three dimensional superintegrable systems in a magnetic field. We study the class of such systems which separate in Cartesian coordinates in the limit when the magnetic field vanishes, i.e. possess two second order integrals of motion of the ‘Cartesian type’. For such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories. We observe that the structure of the leading order terms of the Cartesian type integrals should be considered in a more general form than for the case without magnetic field.

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

“…This is possible only when the coefficient of each power of p 3 vanishes. Namely, when (6), but also the functions u j and V j that allow to find the magnetic field B and potential W as in the more general gauge invariant form (7). In the integrals, L denotes the angular momentum.…”

“…For α 1 = β 1 = ℓ 2 = m 2 = 0 we have the simpler system (16). The case α j = β j = 0, ℓ j = m j = ±1, j = 1, 2 was studied in [7] and it is shown there to be quadratically minimally superintegrable, with the fourth independent integral (besides the two Cartesian ones) inherited from the 2D caged oscillator, of first order. In the more general case (18), the order of the fourth integral can be arbitrarily high, depending on the value of ℓ m .…”

“…The study of superintegrability with magnetic fields was initiated in [1] and subsequently followed in both two spatial dimensions [2,3,4,5] and three spatial dimensions [6,7,8,9,10]. Also a relativistic version of the problem was recently considered, cf.…”

“…In other cases we show that the existence of a quadratic integral necessarily implies the existence of an integral in a particular simpler form, which makes our calculations tractable. When the results of the present paper and [6,7] are viewed together, they provide an exhaustive list of three-dimensional quadratically minimally and maximally superintegrable systems with magnetic fields separable in Cartesian coordinates.…”

“…With these simplifications at hand, the determining equations for the integral can be solved. In section 9.1 we list the superintegrable systems so found; their explicit derivation is rather technical and tedious and we review it in appendices A, B and C. The special case in which the magnetic field is constant and the functions V j in (7) and (9) are at most quadratic polynomials is studied in section 8. Finally, in section 9.2 we discuss the approaches to construction of higher order integrals.…”