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

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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Volume 295, Pages 241–260
DOI: https://doi.org/10.1134/S0371968516040154 (Mi tm3760)

This article is cited in 18 scientific papers (total in 18 papers)

On first integrals of geodesic flows on a two-torus

I. A. Taimanov^ab

^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

Full-text PDF (245 kB) Citations (18)

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

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.

Received: May 26, 2016

English version:
Proceedings of the Steklov Institute of Mathematics, 2016, Volume 295, Pages 225–242
DOI: https://doi.org/10.1134/S0081543816080150

Bibliographic databases:

Document Type: Article

UDC: 517.9+531.01

Language: Russian

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

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Linking options:

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https://doi.org/10.1134/S0371968516040154

https://www.mathnet.ru/eng/tm/v295/p241

This publication is cited in the following 18 articles:

Sergei V. Agapov, Maria V. Demina, “Integrable geodesic flows and metrisable second-order ordinary differential equations”, Journal of Geometry and Physics, 199 (2024), 105168
Sergei Agapov, Vladislav Shubin, “New examples of non-polynomial integrals of two-dimensional geodesic flows *”, J. Phys. A: Math. Theor., 57:1 (2024), 015204
JOSCHA HENHEIK, “Deformational rigidity of integrable metrics on the torus”, Ergod. Th. Dynam. Sys., 2024, 1
S. V. Agapov, Zh. Sh. Fakhriddinov, “O nekotorykh svoistvakh polugamiltonovykh sistem, voznikayuschikh v zadache ob integriruemykh geodezicheskikh potokakh na dvumernom tore”, Sib. matem. zhurn., 64:5 (2023), 881–894
S. V. Agapov, Zh. Sh. Fakhriddinov, “On Some Properties of Semi-Hamiltonian Systems Arising in the Problem of Integrable Geodesic Flows on the Two-Dimensional Torus”, Sib Math J, 64:5 (2023), 1063
Sergei Agapov, Alexey Potashnikov, Vladislav Shubin, “Integrable magnetic geodesic flows on 2-surfaces *”, Nonlinearity, 36:4 (2023), 2128
Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Gyroscopic Chaplygin Systems and Integrable Magnetic Flows on Spheres”, J Nonlinear Sci, 33:3 (2023)
A.I. Klevin, A.V. Tsvetkova, “Nonlinear Long Standing Waves with Support Bounded by Caustics or Localized in the Vicinity of a Two-Link Trajectory”, Russ. J. Math. Phys., 30:4 (2023), 543
Valery V. Kozlov, “On the Integrability of Circulatory Systems”, Regul. Chaotic Dyn., 27:1 (2022), 11–17
V. V. Kozlov, “On the integrability of the equations of dynamics in a non-potential force field”, Russian Math. Surveys, 77:6 (2022), 1087–1106
S. V. Agapov, A. A. Valyuzhenich, V. V. Shubin, “Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torus”, Siberian Math. J., 62:4 (2021), 581–585
O. I. Morozov, J.-H. Chang, “The dispersionless veselov-novikov equation: symmetries, exact solutions, and conservation laws”, Anal. Math. Phys., 11:3 (2021), 126
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Nonstandard Liouville tori and caustics in asymptotics in the form of Airy and Bessel functions for two-dimensional standing coastal waves”, St. Petersburg Math. J., 33:2 (2022), 185–205
S. V. Agapov, “Rational integrals of a natural mechanical system on the 2-torus”, Siberian Math. J., 61:2 (2020), 199–207
S. V. Agapov, “On first integrals of two-dimensional geodesic flows”, Siberian Math. J., 61:4 (2020), 563–574
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “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
S. Agapov, A. Valyuzhenich, “Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels”, Discret. Contin. Dyn. Syst., 39:11 (2019), 6565–6583
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Труды Математического института имени В. А. Стеклова

Proceedings of the Steklov Institute of Mathematics

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