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Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: integrability, superintegrability, higher-order integrals, magnetic field. 
@article{SIGMA_2020_16_a14,
     author = {Antonella Marchesiello and Libor \v{S}nobl},
     title = {Classical {Superintegrable} {Systems} in a {Magnetic} {Field} that {Separate} in {Cartesian} {Coordinates}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a14/}
}
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Antonella Marchesiello; Libor Šnobl. Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a14/

[1] Benenti S., Chanu C., Rastelli G., “Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds”, J. Math. Phys., 42 (2001), 2065–2091 | DOI | MR | Zbl

[2] Bertrand S., Šnobl L., “On rotationally invariant integrable and superintegrable classical systems in magnetic fields with non-subgroup type integrals”, J. Phys. A: Math. Theor., 52 (2019), 195201, 25 pp., arXiv: 1812.09399 | DOI | MR

[3] Bérubé J., Winternitz P., “Integrable and superintegrable quantum systems in a magnetic field”, J. Math. Phys., 45 (2004), 1959–1973, arXiv: math-ph/0311051 | DOI | MR

[4] Charest F., Hudon C., Winternitz P., “Quasiseparation of variables in the Schrödinger equation with a magnetic field”, J. Math. Phys., 48 (2007), 012105, 16 pp., arXiv: math-ph/0502046 | DOI | MR | Zbl

[5] Dorizzi B., Grammaticos B., Ramani A., Winternitz P., “Integrable Hamiltonian systems with velocity-dependent potentials”, J. Math. Phys., 26 (1985), 3070–3079 | DOI | MR | Zbl

[6] Escobar-Ruiz A. M., López Vieyra J. C., Winternitz P., “Fourth order superintegrable systems separating in polar coordinates. I Exotic potentials”, J. Phys. A: Math. Theor., 50 (2017), 495206, 35 pp., arXiv: 1706.08655 | DOI | MR | Zbl

[7] Escobar-Ruiz A. M., Winternitz P., Yurduşen I., “General $N$th-order superintegrable systems separating in polar coordinates”, J. Phys. A: Math. Theor., 51 (2018), 40LT01, 12 pp., arXiv: 1806.06849 | DOI | MR | Zbl

[8] Escobar-Ruiz M. A., Turbiner A. V., “Two charges on a plane in a magnetic field: special trajectories”, J. Math. Phys., 54 (2013), 022901, 15 pp., arXiv: 1208.2995 | DOI | MR | Zbl

[9] Evans N. W., “Superintegrability in classical mechanics”, Phys. Rev. A, 41 (1990), 5666–5676 | DOI | MR

[10] Evans N. W., Verrier P. E., “Superintegrability of the caged anisotropic oscillator”, J. Math. Phys., 49 (2008), 092902, 10 pp., arXiv: 0808.2146 | DOI | MR | Zbl

[11] Heinzl T., Ilderton A., “Superintegrable relativistic systems in spacetime-dependent background fields”, J. Phys. A: Math. Theor., 50 (2017), 345204, 14 pp., arXiv: 1701.09168 | DOI | MR | Zbl

[12] Kotkin G. L., Serbo V. G., Collection of problems in classical mechanics, International Series in Natural Philosophy, 31, Pergamon Press, 1971 | DOI | MR

[13] Makarov A. A., Smorodinsky J. A., Valiev K., Winternitz P., “A systematic search for nonrelativistic systems with dynamical symmetries”, Nuovo Cimento A, 52 (1967), 1061–1084 | DOI

[14] Marchesiello A., Šnobl L., “Superintegrable 3D systems in a magnetic field corresponding to Cartesian separation of variables”, J. Phys. A: Math. Theor., 50 (2017), 245202, 24 pp. | DOI | MR | Zbl

[15] Marchesiello A., Šnobl L., “An infinite family of maximally superintegrable systems in a magnetic field with higher order integrals”, SIGMA, 14 (2018), 092, 11 pp., arXiv: 1804.03039 | DOI | MR | Zbl

[16] Marchesiello A., Šnobl L., Winternitz P., “Three-dimensional superintegrable systems in a static electromagnetic field”, J. Phys. A: Math. Theor., 48 (2015), 395206, 24 pp., arXiv: 1507.04632 | DOI | MR | Zbl

[17] Marchesiello A., Šnobl L., Winternitz P., “Spherical type integrable classical systems in a magnetic field”, J. Phys. A: Math. Theor., 51 (2018), 135205, 24 pp. | DOI | MR | Zbl

[18] Marquette I., “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I Rational function potentials”, J. Math. Phys., 50 (2009), 012101, 23 pp., arXiv: 0807.2858 | DOI | MR | Zbl

[19] Marquette I., “Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. II Painlevé transcendent potentials”, J. Math. Phys., 50 (2009), 095202, 18 pp., arXiv: 0811.1568 | DOI | MR | Zbl

[20] Marquette I., Sajedi M., Winternitz P., “Fourth order superintegrable systems separating in Cartesian coordinates I Exotic quantum potentials”, J. Phys. A: Math. Theor., 50 (2017), 315201, 29 pp., arXiv: 1703.09751 | DOI | MR | Zbl

[21] Marquette I., Sajedi M., Winternitz P., “Two-dimensional superintegrable systems from operator algebras in one dimension”, J. Phys. A: Math. Theor., 52 (2019), 115202, 27 pp., arXiv: 1810.05793 | DOI

[22] Miller Jr. W., Post S., Winternitz P., “Classical and quantum superintegrability with applications”, J. Phys. A: Math. Theor., 46 (2013), 423001, 97 pp., arXiv: 1309.2694 | DOI | MR | Zbl

[23] Post S., Winternitz P., “General $N$th order integrals of motion in the Euclidean plane”, J. Phys. A: Math. Theor., 48 (2015), 405201, 24 pp., arXiv: 1501.00471 | DOI | MR | Zbl

[24] Pucacco G., “On integrable Hamiltonians with velocity dependent potentials”, Celestial Mech. Dynam. Astronom., 90 (2004), 111–125 | DOI | MR | Zbl

[25] Pucacco G., Rosquist K., “Integrable Hamiltonian systems with vector potentials”, J. Math. Phys., 46 (2005), 012701, 25 pp., arXiv: nlin.SI/0405065 | DOI | MR | Zbl

[26] Rodríguez M. A., Tempesta P., Winternitz P., “Reduction of superintegrable systems: the anisotropic harmonic oscillator”, Phys. Rev. E, 78 (2008), 046608, 6 pp., arXiv: 0807.1047 | DOI | MR

[27] Shapovalov V. N., Bagrov V. G., Meshkov A. G., “Separation of variables in the stationary Schrödinger equation”, Soviet Phys. J., 15 (1972), 1115–1119 | DOI | MR

[28] Taut M., “Two electrons in a homogeneous magnetic field: particular analytical solutions”, J. Phys. A: Math. Gen., 27 (1994), 1045–1055 | DOI | MR

[29] Taut M., “Two particles with opposite charge in a homogeneous magnetic field: particular analytical solutions of the two-dimensional Schrödinger equation”, J. Phys. A: Math. Gen., 32 (1999), 5509–5515 | DOI | MR | Zbl

[30] Turbiner A. V., Escobar-Ruiz M. A., “Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions”, J. Phys. A: Math. Theor., 46 (2013), 295204, 15 pp., arXiv: 1303.2345 | DOI | MR | Zbl