Superintegrability of separable systems with magnetic field: the cylindrical case

Abstract

We present a general method simplifying the search for additional integrals of motion of three dimensional systems with magnetic fields. The method is suitable for systems possessing at least one conserved canonical momentum in a suitable coordinates system. It reduces the problem either to consideration of lower dimensional systems or of particular constrained forms of the hypothetical integral. In particular, it is applicable to all separable systems in the Euclidean space since they are known to possess at least one cyclic coordinates when magnetic field is present.Next, we focus on systems which separate in the cylindrical coordinates. Using our method, we are able to classify all superintegrable systems of this kind under the assumption that all considered integrals are at most second order in the momenta. In addition to already known systems, several new minimally superintegrable systems are found and we show that no quadratically maximally superintegrable ones can exist. We also construct some examples of systems with higher order integrals.

Quantum cylindrical integrability in magnetic fields

“…It is therefore probable that some superintegrable systems can be found by imposing further restrictions. This has been done in classical mechanics for separable systems [19] and on the intersection with other integrable systems [20], but only for first order integrals in quantum mechanics [16]. Easing these restrictions is necessary as well as going beyond integrals connected to orthogonal separation of variables as was shown in [14].…”

Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

“…The first systematic attempt to study the systems with vector potentials seems to be [10], followed by several papers investigating integrable and superintegrable systems with magnetic fields in two spatial dimensions, see [11][12][13]. In three spatial dimensions two of the present authors (AM and L Š) together with Winternitz started to address the problem in [14] and then followed this line of research with various collaborators in [15][16][17][18][19][20][21]. In several of these papers we have seen that the one-to-one relation between integrability with integrals at most quadratic in the momenta and separability of the Hamilton-Jacobi equation in the configuration space (which was established for natural systems in [2] by direct comparison of the obtained systems with the results of Eisenhart on separability [22,23]) no longer holds in the presence of magnetic fields.…”

Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

“…Also, three dimensional quadratically integrable systems in magnetic fields which possess non-subgroup type quadratic integrals have been studied [26]. Superintegrability and separability of the systems in magnetic field with the integrals at most quadratic in the momenta which admit at least one cyclic coordinate, were investigated in [27].…”