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Mat. Sb., 1994, Volume 185, Number 12, Pages 49–64 (Mi msb946)  

This article is cited in 17 scientific papers (total in 17 papers)

Polynomial integrals of geodesic flows on a two-dimensional torus

V. V. Kozlov, N. V. Denisova

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The geodesic curves of a Riemannian metric on a surface are described by a Hamiltonian system with two degrees of freedom whose Hamiltonian is quadratic in the momenta. Because of the homogeneity, every integral of the geodesic problem is a function of integrals that are polynomial in the momenta. The geodesic flow on a surface of genus greater than one does not admit an additional nonconstant integral at all, but on the other hand there are numerous examples of metrics on a torus whose geodesic flows are completely integrable: there are polynomial integrals of degree $\leqslant2$ that are independent of the Hamiltonian. It appears that the degree of an additional 'irreducible' polynomial integral of a geodesic flow on a torus cannot exceed two. In the present paper this conjecture is proved for metrics which can arbitrarily closely approximate any metric on a two-dimensional torus.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 83:2, 469–481

Bibliographic databases:

Document Type: Article
UDC: 517.9+531.01
MSC: Primary 58F17, 58F05; Secondary 70M05, 15A24, 05A19
Received: 07.04.1994

Citation: V. V. Kozlov, N. V. Denisova, “Polynomial integrals of geodesic flows on a two-dimensional torus”, Mat. Sb., 185:12 (1994), 49–64; Russian Acad. Sci. Sb. Math., 83:2 (1995), 469–481

Citation in format AMSBIB
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\by V.~V.~Kozlov, N.~V.~Denisova
\paper Polynomial integrals of geodesic flows on a~two-dimensional torus
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 12
\pages 49--64
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1317298}
\zmath{https://zbmath.org/?q=an:0841.53039}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 83
\issue 2
\pages 469--481
\crossref{https://doi.org/10.1070/SM1995v083n02ABEH003601}
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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. A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko, “The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body”, Russian Math. Surveys, 50:3 (1995), 473–501  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. N. V. Denisova, “The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus”, Sb. Math., 188:7 (1997), 1055–1069  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Sb. Math., 189:10 (1998), 1441–1466  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. N. V. Denisova, “Integrals polynomial in velocity for two-degrees-of-freedom dynamical systems whose configuration space is a torus”, Math. Notes, 64:1 (1998), 31–37  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Denisova N., “Polynomial Fields of the Third Degree Symmetries of Geodesic Flows on a Two-Dimensional Torus”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1998, no. 2, 48–53  mathscinet  zmath  isi
    6. N. V. Denisova, V. V. Kozlov, “Polynomial integrals of reversible mechanical systems with a two-dimensional torus as the configuration space”, Sb. Math., 191:2 (2000), 189–208  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. Kozlova T., “Billiard Systems with Polynomial Integrals of Third and Fourth Degree”, J. Phys. A-Math. Gen., 34:11 (2001), 2121–2124  crossref  mathscinet  zmath  adsnasa  isi
    8. Kozlov V., “Topological Obstructions to the Existence of Quantum Conservation Laws”, Dokl. Math., 71:2 (2005), 300–302  mathnet  mathscinet  zmath  isi  elib
    9. Bialy M., Mironov A.E., “Cubic and Quartic Integrals for Geodesic Flow on 2-Torus via a System of the Hydrodynamic Type”, Nonlinearity, 24:12 (2011), 3541–3554  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. N. V. Denisova, V. V. Kozlov, D. V. Treschev, “Remarks on polynomial integrals of higher degrees for reversible systems with toral configuration space”, Izv. Math., 76:5 (2012), 907–921  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    11. S. V. Agapov, “Ob integriruemom geodezicheskom potoke v magnitnom pole na dvumernom tore”, Sib. elektron. matem. izv., 12 (2015), 868–873  mathnet  crossref
    12. V. S. Kalnitsky, “Symmetries of a flat cosymbol algebra of the differential operators”, J. Math. Sci. (N. Y.), 222:4 (2017), 429–436  mathnet  crossref  mathscinet
    13. I. A. Taimanov, “On first integrals of geodesic flows on a two-torus”, Proc. Steklov Inst. Math., 295 (2016), 225–242  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. Pavlov M.V. Tsarev S.P., “On local description of two-dimensional geodesic flows with a polynomial first integral”, J. Phys. A-Math. Theor., 49:17 (2016), 175201  crossref  mathscinet  zmath  isi  scopus
    15. Thierry Combot, “Rational Integrability of Trigonometric Polynomial Potentials on the Flat Torus”, Regul. Chaotic Dyn., 22:4 (2017), 386–497  mathnet  crossref
    16. Heil K. Moroianu A. Semmelmann U., “Killing Tensors on Tori”, J. Geom. Phys., 117 (2017), 1–6  crossref  mathscinet  zmath  isi  scopus
    17. Bolsinov A. Matveev V.S. Miranda E. Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430  crossref  isi  scopus
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