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Izv. Akad. Nauk SSSR Ser. Mat., 1982, Volume 46, Issue 5, Pages 994–1010 (Mi izv1656)  

This article is cited in 46 scientific papers (total in 46 papers)

Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities

V. N. Kolokoltsov


Abstract: In the paper an explicit description is given for all Riemannian metrics on the sphere and on the torus whose geodesic flows have an additional first integral that is both quadratic in the velocities and independent of the energy integral. Moreover, it is proved that on compact two-dimensional manifolds of higher genus the geodesic flows have no additional polynomial integral. All the results admit straightforward generalizations to arbitrary natural systems given on cotangent bundles of two-dimensional manifolds.
Bibliography: 8 titles.

Full text: PDF file (1672 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1983, 21:2, 291–306

Bibliographic databases:

UDC: 513.88
MSC: Primary 58F17, 53C22; Secondary 34C35, 58F07
Received: 15.02.1982

Citation: V. N. Kolokoltsov, “Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities”, Izv. Akad. Nauk SSSR Ser. Mat., 46:5 (1982), 994–1010; Math. USSR-Izv., 21:2 (1983), 291–306

Citation in format AMSBIB
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\paper Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial in the velocities
\jour Izv. Akad. Nauk SSSR Ser. Mat.
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\jour Math. USSR-Izv.
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\pages 291--306
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. V. Kozlov, “Integrability and non-integrability in Hamiltonian mechanics”, Russian Math. Surveys, 38:1 (1983), 1–76  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Russian Math. Surveys, 39:2 (1984), 1–67  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658  mathnet  crossref  mathscinet  zmath
    4. I. A. Taimanov, “Topological obstructions to integrability of geodesic flows on non-simply-connected manifolds”, Math. USSR-Izv., 30:2 (1988), 403–409  mathnet  crossref  mathscinet  zmath  isi
    5. M. L. Byalyi, “First integrals that are polynomial in momenta for a mechanical system on a two-dimensional torus”, Funct. Anal. Appl., 21:4 (1987), 310–312  mathnet  crossref  mathscinet  zmath  isi
    6. E. N. Selivanova, “Classification of geodesic flows of Liouville metrics on the two-dimensional torus up to topological equivalence”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 491–505  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. Nguyen Tien Zung, “The complexity of integrable Hamiltonian systems on a prescribed three-dimensional constant-energy submanifold”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 507–533  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. V. V. Kozlov, N. V. Denisova, “Symmetries and the topology of dynamical systems with two degrees of freedom”, Russian Acad. Sci. Sb. Math., 80:1 (1995), 105–124  mathnet  crossref  mathscinet  zmath  isi
    9. Nguyen Tien Zung, L. S. Polyakova, E. N. Selivanova, “Topological Classification of Integrable Geodesic Flows on Orientable Two-Dimensional Riemannian Manifolds with Additional Integral Depending on Momenta Linearly or Quadratically”, Funct. Anal. Appl., 27:3 (1993), 186–196  mathnet  crossref  mathscinet  zmath  isi
    10. V. V. Kozlov, N. V. Denisova, “Polynomial integrals of geodesic flows on a two-dimensional torus”, Russian Acad. Sci. Sb. Math., 83:2 (1995), 469–481  mathnet  crossref  mathscinet  zmath  isi
    11. A. V. Bolsinov, A. T. Fomenko, “Integrable geodesic flows on the sphere, generated by Goryachev–Chaplygin and Kowalewski systems in the dynamics of a rigid body”, Math. Notes, 56:2 (1994), 859–861  mathnet  crossref  mathscinet  zmath  isi
    12. K Rosquist, G Pucacco, J Phys A Math Gen, 28:11 (1995), 3235  crossref  mathscinet  zmath  adsnasa
    13. I. K. Babenko, N. N. Nekhoroshev, “On complex structures on two-dimensional tori admitting metrics with nontrivial quadratic integral”, Math. Notes, 58:5 (1995), 1129–1135  mathnet  crossref  mathscinet  zmath  isi
    14. V. V. Kalashnikov, “Topological classification of quadratic-integrable geodesic flows on a two-dimensional torus”, Russian Math. Surveys, 50:1 (1995), 200–201  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    15. 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
    16. V. V. Kozlov, V. V. Ten, “Topology of domains of possible motions of integrable systems”, Sb. Math., 187:5 (1996), 679–684  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. Ya. B. Vorobets, “Asymptotics of the spectrum of the Laplace–Beltrami operator on tori with Liouville and infra-Liouville metrics”, Russian Math. Surveys, 52:2 (1997), 430–431  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    18. 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
    19. 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
    20. 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
    21. S. Yu. Dobrokhotov, A. I. Shafarevich, “Tunnel Splitting of the Spectrum of the Beltrami–Laplace Operators on Two-Dimensional Surfaces with Square Integrable Geodesic Flow”, Funct. Anal. Appl., 34:2 (2000), 133–134  mathnet  crossref  crossref  mathscinet  zmath  isi
    22. 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
    23. A. V. Bolsinov, I. A. Taimanov, “Integrable Geodesic Flows on the Suspensions of Toric Automorphisms”, Proc. Steklov Inst. Math., 231 (2000), 42–58  mathnet  mathscinet  zmath
    24. A. V. Bolsinov, B. Jovanović, “Integrable geodesic flows on homogeneous spaces”, Sb. Math., 192:7 (2001), 951–968  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. Max Karlovini, Giuseppe Pucacco, Kjell Rosquist, Lars Samuelsson, “A unified treatment of quartic invariants at fixed and arbitrary energy”, J Math Phys (N Y ), 43:8 (2002), 4041  crossref  mathscinet  zmath  isi
    26. Vladimir S. Matveev, “Three-dimensional manifolds having metrics with the same geodesics”, Topology, 42:6 (2003), 1371  crossref
    27. Bolsinov A.V., “Integrable geodesic flows on Riemannian manifolds: Construction and obstructions”, Proceedings of the Workshop on Contemporary Geometry and Related Topics, 2004, 57–103  isi
    28. Giuseppe Pucacco, Kjell Rosquist, “Configurational invariants of Hamiltonian systems”, J Math Phys (N Y ), 46:5 (2005), 052902  crossref  mathscinet  zmath  isi
    29. Alexey V. Bolsinov, Vladimir S. Matveev, Giuseppe Pucacco, “Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta”, Journal of Geometry and Physics, 59:7 (2009), 1048  crossref
    30. GIUSEPPE PUCACCO, KJELL ROSQUIST, “NONSTANDARD SEPARABILITY ON THE Minkowski PLANE”, J. Nonlinear Math. Phys, 16:04 (2009), 421  crossref
    31. Vladimir S. Matveev, Vsevolod V. Shevchishin, “Differential invariants for cubic integrals of geodesic flows on surfaces”, Journal of Geometry and Physics, 60:6-8 (2010), 833  crossref  elib
    32. V. T. Lisitsa, “On the conditions of total resonance of Liouville type Hamiltonian systems with $n$ degrees of freedom”, Zhurn. matem. fiz., anal., geom., 6:3 (2010), 295–304  mathnet  mathscinet  zmath  elib
    33. Vladimir S. Matveev, Vsevolod V. Shevchishin, “Two-dimensional superintegrable metrics with one linear and one cubic integral”, Journal of Geometry and Physics, 61:8 (2011), 1353  crossref
    34. Vladimir S. Matveev, “Two-dimensional metrics admitting precisely one projective vector field”, Math. Ann, 2011  crossref
    35. Thomas J. Waters, “Regular and irregular geodesics on spherical harmonic surfaces”, Physica D: Nonlinear Phenomena, 2011  crossref
    36. Misha Bialy, A.E.. Mironov, “Integrable geodesic flows on 2-torus: Formal solutions and variational principle”, Journal of Geometry and Physics, 2014  crossref
    37. Giuseppe Pucacco, “Polynomial separable indefinite natural systems”, Journal of Geometry and Physics, 2014  crossref
    38. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    39. V. A. Sharafutdinov, “Killing tensor fields on the $2$-torus”, Siberian Math. J., 57:1 (2016), 155–173  mathnet  crossref  crossref  mathscinet  isi  elib
    40. V. V. Kozlov, D. V. Treschev, “Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics”, Sb. Math., 207:10 (2016), 1435–1449  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    41. 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
    42. 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
    43. 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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    46. S. V. Agapov, “O pervykh integralakh dvumernykh geodezicheskikh potokov”, Sib. matem. zhurn., 61:4 (2020), 721–734  mathnet  crossref
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