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 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 Regul. Chaotic Dyn., 2015, Volume 20, Issue 6, Pages 667–678 (Mi rcd36)

On an Integrable Magnetic Geodesic Flow on the Two-torus

Iskander A. Taimanovab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Department of Mechanics and Mathematics, Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia

Abstract: The magnetic geodesic flow on a flat two-torus with the magnetic field $F=\cos(x)dx\wedge dy$ is completely integrated and the description of all contractible periodic magnetic geodesics is given. It is shown that there are no such geodesics for energy $E\geqslant1/2$, for $E<1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.

Keywords: integrable system, magnetic geodesic flow

 Funding Agency Grant Number Russian Science Foundation 14-11-00441 The work was supported by RSF (grant 14-11-00441).

DOI: https://doi.org/10.1134/S1560354715060039

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Bibliographic databases:

Document Type: Article
MSC: 53D25, 37J35
Accepted:20.10.2015
Language: English

Citation: Iskander A. Taimanov, “On an Integrable Magnetic Geodesic Flow on the Two-torus”, Regul. Chaotic Dyn., 20:6 (2015), 667–678

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242
2. S. V. Bolotin, V. V. Kozlov, “Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom”, Izv. Math., 81:4 (2017), 671–687
3. L. Asselle, G. Benedetti, “On the periodic motions of a charged particle in an oscillating magnetic field on the two-torus”, Math. Z., 286:3-4 (2017), 843–859
4. S. Chanda, G. W. Gibbons, P. Guha, “Jacobi–Maupertuis metric and Kepler equation”, Int. J. Geom. Methods Mod. Phys., 14:7 (2017), 1730002
5. S. V. Agapov, M. Bialy, A. E. Mironov, “Integrable magnetic geodesic flows on 2-torus: new examples via quasi-linear system of PDEs”, Commun. Math. Phys., 351:3 (2017), 993–1007