Three-dimensional superintegrable systems in a static electromagnetic field

Abstract

We consider a charged particle moving in a static electromagnetic field described by the vector potential A( x) and the electrostatic potential V ( x). We study the conditions on the structure of the integrals of motion of the first and second order in momenta, in particular how they are influenced by the gauge invariance of the problem. Next, we concentrate on the three possibilities for integrability arising from the first order integrals corresponding to three nonequivalent subalgebras of the Euclidean algebra, namely (P 1 , P 2 ), (L 3 , P 3 ) and (L 1 , L 2 , L 3 ). For these cases we look for additional independent integrals of first or second order in the momenta. These would make the system superintegrable (minimally or maximally). We study their quantum spectra and classical equations of motion. In some cases nonpolynomial integrals of motion occur and ensure maximal superintegrability.

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.…”

“…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.…”

“…Minimal superintegrability due to the existence of another first order integral has been studied in [6,7]. Here we investigate the conditions for the existence of an additional integral of order at least two for the systems (7), (9).…”

“…Quadratic superintegrability in Case IWe start with the class of systems in(7). To fix the ideas, let us choose a gauge as in(6) and assume that there exists a quadratic independent integral I. Thus, by Proposition 3 there exist another quadratic integral X such that {X, p 3 } is at most first order as a polynomial in the momenta.…”

Cylindrical type integrable classical systems in a magnetic field

“…= 0 (8) using the coefficients in front of each individual combination of powers in momenta. Those equations in cartesian coordinates are listed in previous papers [36][37][38][39][40].…”