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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.

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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
\by V.~N.~Kolokoltsov
\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.
\yr 1982
\vol 46
\issue 5
\pages 994--1010
\jour Math. USSR-Izv.
\yr 1983
\vol 21
\issue 2
\pages 291--306

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    This publication is cited in the following articles:
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    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
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