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 Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 241–260 (Mi tm3760)

On first integrals of geodesic flows on a two-torus

I. A. Taimanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090 Russia
b Faculty of Mechanics and Mathematics, Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia

Abstract: For a geodesic (or magnetic geodesic) flow, the problem of the existence of an additional (independent of the energy) first integral that is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.

 Funding Agency Grant Number Russian Science Foundation 14-11-00441 This work is supported by the Russian Science Foundation under grant 14-11-00441.

DOI: https://doi.org/10.1134/S0371968516040154

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 225–242

Bibliographic databases:

UDC: 517.9+531.01

Citation: I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 241–260; Proc. Steklov Inst. Math., 295 (2016), 225–242

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3760
• https://doi.org/10.1134/S0371968516040154
• http://mi.mathnet.ru/eng/tm/v295/p241

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This publication is cited in the following articles:
1. 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
2. Anikin A.Yu. Dobrokhotov S.Yu. Nazaikinskii V.E. Tsvetkova A.V., “Asymptotic Eigenfunctions of the Operator Delta D(X)Delta Defined in a Two-Dimensional Domain and Degenerating on Its Boundary and Billiards With Semi-Rigid Walls”, Differ. Equ., 55:5 (2019), 644–657
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