Abstract:
The magnetic geodesic flow on a flat two-torus with the magnetic field F=cos(x)dx∧dyF=cos(x)dx∧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⩾1/2, for E<1/2 simple periodic magnetic geodesics form two S1-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.
\Bibitem{Tai15}
\by Iskander A. Taimanov
\paper On an Integrable Magnetic Geodesic Flow on the Two-torus
\jour Regul. Chaotic Dyn.
\yr 2015
\vol 20
\issue 6
\pages 667--678
\mathnet{http://mi.mathnet.ru/rcd36}
\crossref{https://doi.org/10.1134/S1560354715060039}
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https://www.mathnet.ru/eng/rcd36
https://www.mathnet.ru/eng/rcd/v20/i6/p667
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Gabriela P. Ovando, Mauro Subils, “Magnetic Trajectories on 2-Step Nilmanifolds”, J Geom Anal, 33:6 (2023)
Sergei Agapov, Alexey Potashnikov, Vladislav Shubin, “Integrable magnetic geodesic flows on 2-surfaces
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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
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
S. Chanda, G. W. Gibbons, P. Guha, “Jacobi–Maupertuis metric and Kepler equation”, Int. J. Geom. Methods Mod. Phys., 14:7 (2017), 1730002
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
I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242
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