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2013
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Iskander Taimanov

Iskander Taimanov
4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia
Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences

D. Sc., Professor

Corresponding Member of RAS

Author and editor in ICS (ICS series «Contemporary Mathematics», «Computing in Mathematics, Physics and Biology»)

Member of the editorial board of «Regular & Chaotic Dynamics»

Publications:

Taimanov I. A.
On an Integrable Magnetic Geodesic Flow on the Two-torus
2015, vol. 20, no. 6, pp.  667-678
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 $E \geqslant 1/2$, 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
Citation: Taimanov I. A.,  On an Integrable Magnetic Geodesic Flow on the Two-torus, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 667-678
DOI:10.1134/S1560354715060039
Taimanov I. A.
Periodic magnetic geodesics on almost every energy level via variational methods
2010, vol. 15, no. 4-5, pp.  598-605
Abstract
For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy level the principle of throwing out cycles gives periodic magnetic geodesics on the critical levels defined by the "thrown out" cycles.
Keywords: magnetic geodesics, closed extremals, calculus of variations in the large
Citation: Taimanov I. A.,  Periodic magnetic geodesics on almost every energy level via variational methods, Regular and Chaotic Dynamics, 2010, vol. 15, no. 4-5, pp. 598-605
DOI:10.1134/S1560354710040131
Taimanov I. A.
The type numbers of closed geodesics
2010, vol. 15, no. 1, pp.  84-100
Abstract
This is a short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics.
Keywords: closed geodesic, Morse theory, loop space, Finsler metric
Citation: Taimanov I. A.,  The type numbers of closed geodesics, Regular and Chaotic Dynamics, 2010, vol. 15, no. 1, pp. 84-100
DOI:10.1134/S1560354710010053

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