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

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 3-dimensional 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 4-potential 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.

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