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

This article is cited in 5 scientific papers (total in 5 papers)

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

References: PDF file   HTML file

Bibliographic databases:

Document Type: Article
MSC: 53D25, 37J35
Received: 15.08.2015
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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\paper On an Integrable Magnetic Geodesic Flow on the Two-torus
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\vol 20
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\pages 667--678
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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  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    4. S. Chanda, G. W. Gibbons, P. Guha, “Jacobi–Maupertuis metric and Kepler equation”, Int. J. Geom. Methods Mod. Phys., 14:7 (2017), 1730002  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  isi  scopus
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