The algebra of the integrals of motion, linear in the momenta, of magnetic geodesic flows on arbitrary Riemannian manifolds is investigated. It is shown that the indicated algebra is a one-dimensional central extension of the Lie algebra of Killing vector fields of the manifold which conserve the external field. A constructive method is proposed for performing an integration in quadratures of the magnetic geodesic flows on homogeneous Riemannian manifolds of zero index.
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References
L. D. Landau and E. M. Lifshitz, Theoretical Physics. Vol. 2. The Classical Theory of Fields. Fourth Edition, Butterworth Heinemann, Oxford, UK (1975).
S. P. Novikov, Usp. Matem. Nauk, 37, 3–49 (1982).
J. W. Van Holten, Phys. Rev. Lett., 75, 1–9 (2007).
A. V. Bolsinov and B. Jovanovic, Comm. Math. Helv., 83, 679–700 (2008).
I. V. Shirokov, Teor. Mat. Fiz., 126, No. 3, 393–408 (2001).
A. A. Magazev and I. V. Shirokov, Teor. Mat. Fiz., 136, No. 3, 365–379 (2003).
A. A. Magazev, I. V. Shirokov, and Yu. A. Yurevich, Teor. Mat. Fiz., 156, No. 2, 189–206 (2008).
M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization [in Russian], Nauka, Moscow (1991).
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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 26–32, March, 2014.
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Magazev, A.A. Magnetic Geodesic Flows on Homogeneous Manifolds. Russ Phys J 57, 312–320 (2014). https://doi.org/10.1007/s11182-014-0241-7
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DOI: https://doi.org/10.1007/s11182-014-0241-7