Geodesic
An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We construct an additional independent integral of motion for a class of three dimensional minimally superintegrable systems with constant magnetic field. This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages] and it is known to possess periodic closed orbits. In the present paper we demonstrate that it is maximally superintegrable. Depending on the values of the parameters of the system, the newly found integral can be of arbitrarily high polynomial order in momenta.
Keywords: integrability; superintegrability; higher order integrals; magnetic field. 
@article{SIGMA_2018_14_a91,
     author = {Antonella Marchesiello and Libor \v{S}nobl},
     title = {An {Infinite} {Family} of {Maximally} {Superintegrable} {Systems~in~a~Magnetic} {Field} with {Higher} {Order} {Integrals}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a91/}
}
TY  - JOUR
AU  - Antonella Marchesiello
AU  - Libor Šnobl
TI  - An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a91/
LA  - en
ID  - SIGMA_2018_14_a91
ER  - 
%0 Journal Article
%A Antonella Marchesiello
%A Libor Šnobl
%T An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a91/
%G en
%F SIGMA_2018_14_a91
Antonella Marchesiello; Libor Šnobl. An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a91/

[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] 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 | Zbl

[3] Dehin D., Hussin V., “A simple connection between the motion in a constant magnetic field and the harmonic oscillator”, Helv. Phys. Acta, 60 (1987), 552–559 | DOI

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

[5] 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

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

[7] Evans N. W., “Group theory of the Smorodinsky–Winternitz system”, J. Math. Phys., 32 (1991), 3369–3375 | DOI | MR | Zbl

[8] 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

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

[10] Jauch J. M., Hill E. L., “On the problem of degeneracy in quantum mechanics”, Phys. Rev., 57 (1940), 641–645 | DOI | MR | Zbl

[11] Labelle S., Mayrand M., Vinet L., “Symmetries and degeneracies of a charged oscillator in the field of a magnetic monopole”, J. Math. Phys., 32 (1991), 1516–1521 | DOI | MR | Zbl

[12] 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

[13] 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

[14] 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

[15] 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

[16] McIntosh H. V., Cisneros A., “Degeneracy in the presence of a magnetic monopole”, J. Math. Phys., 11 (1970), 896–916 | DOI | MR

[17] McSween E., Winternitz P., “Integrable and superintegrable Hamiltonian systems in magnetic fields”, J. Math. Phys., 41 (2000), 2957–2967 | DOI | MR | Zbl

[18] 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

[19] Nakagawa K., Yoshida H., “A list of all integrable two-dimensional homogeneous polynomial potentials with a polynomial integral of order at most four in the momenta”, J. Phys. A: Math. Gen., 34 (2001), 8611–8630, arXiv: nlin.SI/0108051 | DOI | MR | Zbl

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

[21] 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

[22] 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

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

[24] Tempesta P., Turbiner A. V., Winternitz P., “Exact solvability of superintegrable systems”, J. Math. Phys., 42 (2001), 4248–4257, arXiv: hep-th/0011209 | DOI | MR | Zbl

[25] Thirring W., Classical mathematical physics. Dynamical systems and field theories, 3rd ed., Springer-Verlag, New York, 1997 | DOI | MR

[26] 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

[27] Verrier P. E., Evans N. W., “A new superintegrable Hamiltonian”, J. Math. Phys., 49 (2008), 022902, 8 pp., arXiv: 0712.3677 | DOI | MR | Zbl

[28] Zhalij A., “Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case”, J. Phys. Conf. Ser., 621 (2015), 012019, 8 pp., arXiv: 0812.2682 | DOI