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

This article is cited in 7 scientific papers (total in 7 papers)

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

Full text: PDF file (245 kB)
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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 225–242

Bibliographic databases:

UDC: 517.9+531.01
Received: May 26, 2016

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

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. 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  crossref  isi
    3. 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  crossref  mathscinet  zmath  isi
    4. S. V. Agapov, “Ratsionalnye integraly naturalnoi mekhanicheskoi sistemy na dvumernom tore”, Sib. matem. zhurn., 61:2 (2020), 255–265  mathnet  crossref
    5. S. V. Agapov, “O pervykh integralakh dvumernykh geodezicheskikh potokov”, Sib. matem. zhurn., 61:4 (2020), 721–734  mathnet  crossref
    6. A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “Nestandartnye liuvillevy tory i kaustiki v asimptotikakh v vide funktsii Eiri i Besselya dlya dvumernykh stoyachikh beregovykh voln”, Algebra i analiz, 33:2 (2021), 5–34  mathnet
    7. S. V. Agapov, A. A. Valyuzhenich, V. V. Shubin, “Nekotorye zamechaniya o polinomialnykh integralakh vysokoi stepeni magnitnogo geodezicheskogo potoka na dvumernom tore”, Sib. matem. zhurn., 62:4 (2021), 715–720  mathnet  crossref
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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