
SIGMA 6 (2010), 004, 34 pages arXiv:1001.1550
https://doi.org/10.3842/SIGMA.2010.004
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics
Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models
V.V. Kudryashov, Yu.A. Kurochkin, E.M. Ovsiyuk and V.M. Red'kov
Institute of Physics, National Academy of Sciences of Belarus, Minsk, Belarus
Received July 20, 2009, in final form December 29, 2009; Published online January 10, 2010
Abstract
Motion of a classical particle in 3dimensional Lobachevsky and
Riemann spaces is studied in the presence of an external magnetic
field which is analogous to a constant uniform magnetic
field in Euclidean space.
In both cases three integrals of motions are constructed and equations of motion
are solved exactly in the special cylindrical coordinates on the base of the method of separation
of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable,
in Riemann space all motions are finite and periodical.
The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding
4potential description is demonstrated explicitly.
The role of the symmetry is clarified in classification of all
possible solutions, based on the geometric symmetry
group, SO(3,1) and SO(4) respectively.
Key words:
Lobachevsky and Riemann spaces; magnetic field; mechanics in curved space; geometric and gauge symmetry; dynamical systems.
pdf (1110 kb)
ps (413 kb)
tex (782 kb)
References
 Avron J.E., Pnueli A.,
Landau Hamiltonians on symmetric spaces,
in Ideas and Methods in Mathematical Analysis, Stochastics, and Applications (Oslo, 1988),
Cambridge Univ. Press, Cambridge, 1992, 96117.
 Bogush A.A., Red'kov V.M., Krylov G.G.,
Quantum particle in uniform magnetic field on the background of Lobachevsky space,
Dokl. Nats. Akad. Nauk Belarusi 53 (2009), 4551.
 Bogush A.A., Red'kov V.M., Krylov G.G.,
Quantum particle in uniform magnetic field on the background of spherical Riemann space,
Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.Mat. Navuk 2 (2009), 5763.
 Bogush A.A., Red'kov V.M., Krylov G.G.,
Schrödinger particle in magnetic and electric fields in Lobachevsky and Riemann spaces,
Nonlinear Phenom. Complex Syst. 11 (2008), 403416.
 Cappelli A., Dunne G.V., Trugenberger C.A., Zemba G.R.
Conformal symmetry and universal properties of quantum Hall states,
Nuclear Phys. B. 398 (1993), 531567, hepth/9211071.
 Cappelli A., Trugenberger C.A., Zemba G.R.,
Infinite symmetry in the quantum Hall effect,
Nuclear Phys. B 396 (1993), 465490,
hepth/9206027.
 Cariñena J.F., Rañada M.F., Santander M.,
Central potentials on spaces of constant curvature:
the Kepler problem on the twodimensional sphere S^{2} and the hyperbolic plane H^{2},
J. Math. Phys. 46 (2005), 052702, 25 pages,
mathph/0504016.
 Drukker N., Fiol B., Simón J.,
Gödeltype universes and the Landau problem,
J. Cosmol. Astropart. Phys. 2004 (2004), no. 10, 012, 20 pages, hepth/0309199.
 Dunne G.V.,
Hilbert space for charged particles in perpendicular magnetic fields,
Ann. Physics 215 (1992), 233263.
 Gadella M., Negro J., Pronko G.P., Santander M.,
Classical and quantum integrability in 3D system,
J. Phys. A: Math. Theor. 41 (2008), 304030, 15 pages,
arXiv:0711.4915.
 Herranz J., Ballesteros A.,
Superintegrability on threedimensional Riemannian and relativistic spaces of constant curvature,
SIGMA 2 (2006), 010, 22 pages,
mathph/0512084.
 Klauder J.R., Onofri E.,
Landau levels and geometric quantization,
Internat. J. Modern Phys. A 4 (1989), 39393949.
 Kudryashov V.V., Kurochkin Yu.A., Ovsiyuk E.M., Red'kov V.M.,
Motion caused by magnetic field in Lobachevsky space,
in Proceedings of the 1st Zeldovich meeting "The Sun, the Stars, the Universe and General Relativity" (Minsk, April 2023, 2009), Editors R. Ruffini and G. Vereshchagin, to appear.
 Landau L.D.,
Diamagnetismus der Metalle,
Z. f. Physik 64 (1930), 629637.
 Landau L.D., Lifshitz E.M.,
Theory of field, Nauka, Moscow, 1973.
 Landau L.D., Lifshitz E.M.,
Quantum mechanics, nonrelativistic theory, Nauka,
Moscow, 1974.
 Negro J., del Olmo M.A., RodríguezMarco A.,
Landau quantum systems: an approach based on symmetry,
J. Phys. A: Math. Gen. 35 (2002), 22832307,
quantph/0110152.
 Olevsky M.N.,
Threeorthogonal coordinate systems in spaces of constant curvature,
in which equation ΔU + λU = 0 permits the full separation of variables,
Mat. Sb. 27 (1950), 379426.
 Onofri E.,
Landau levels on a torus,
Internat. J. Theoret. Phys. 40 (2001), 537549,
quantph/0007055.

