Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 015, 35 pages      arXiv:1911.01180      https://doi.org/10.3842/SIGMA.2020.015

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

Antonella Marchesiello a and Libor Šnobl b
a) Czech Technical University in Prague, Faculty of Information Technology, Department of Applied Mathematics, Th'akurova 9, 160 00 Prague 6, Czech Republic
b) Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Physics, Bv rehová 7, 115 19 Prague 1, Czech Republic

Received November 05, 2019, in final form March 06, 2020; Published online March 12, 2020

Abstract
We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.

Key words: integrability; superintegrability; higher-order integrals; magnetic field.

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