Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels

Sergei Agapov; Alexandr Valyuzhenich

doi:10.3934/dcds.2019285

Article Contents

2019, Volume 39, Issue 11: 6565-6583. Doi: 10.3934/dcds.2019285

This issue Previous Article Moduli of stability for heteroclinic cycles of periodic solutions Next Article Almost surely invariance principle for non-stationary and random intermittent dynamical systems

Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels

Sergei Agapov^1,2, , and
Alexandr Valyuzhenich^1,

1.
Sobolev Institute of Mathematics, Novosibirsk, 4 Acad. Koptyug avenue, 630090, Russia
2.
Novosibirsk State University, Novosibirsk, 1 Pirogova str., 630090, Russia

^* Corresponding author: Sergei Agapov
^* Corresponding author: Sergei Agapov

Received: December 31, 2018

Published: August 2019

The first author is supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation)

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Abstract

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral $ F $ on $ N+2 $ different energy levels which is polynomial in momenta of an arbitrary degree $ N $ with analytic periodic coefficients. It is proved that in this case the magnetic field and the metric are functions of one variable and there exists a linear in momenta first integral on all energy levels.

Keywords:

Magnetic geodesic flow,
polynomial first integrals.

Mathematics Subject Classification: Primary: 37J35; Secondary: 53D25.

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References

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