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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 6, Pages 601–606 (Mi rcd8)

On Rational Integrals of Geodesic Flows

Valery V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy–Kovalevskaya theorem.

Keywords: conformal coordinates, rational integral, irreducible integrals, Cauchy–Kovalevskaya theorem

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the Russian Scientific Foundation.

DOI: https://doi.org/10.1134/S156035471406001X

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MSC: 34A34, 58E10
Accepted:17.10.2014
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Citation: Valery V. Kozlov, “On Rational Integrals of Geodesic Flows”, Regul. Chaotic Dyn., 19:6 (2014), 601–606

Citation in format AMSBIB
\Bibitem{Koz14} \by Valery V. Kozlov \paper On Rational Integrals of Geodesic Flows \jour Regul. Chaotic Dyn. \yr 2014 \vol 19 \issue 6 \pages 601--606 \mathnet{http://mi.mathnet.ru/rcd8} \crossref{https://doi.org/10.1134/S156035471406001X} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3284603} \zmath{https://zbmath.org/?q=an:06507821} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2014RCD....19..601K} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000345996200001} 

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This publication is cited in the following articles:
1. S. P. Kuznetsov, “On the validity of the nonholonomic model of the rattleback”, Phys. Usp., 58:12 (2015), 1223–1224
2. A. V. Borisov, I. S. Mamaev, “Zamechaniya o novykh modelyakh treniya i negolonomnoi mekhanike”, UFN, 185:12 (2015), 1339–1341
3. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453
4. Alain Albouy, “Projective Dynamics and First Integrals”, Regul. Chaotic Dyn., 20:3 (2015), 247–276
5. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290
6. Valery V. Kozlov, “On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature”, Regul. Chaotic Dyn., 21:7-8 (2016), 821–831
7. A. Aoki, T. Houri, K. Tomoda, “Rational first integrals of geodesic equations and generalised hidden symmetries”, Classical Quantum Gravity, 33:19 (2016), 195003, 12 pp.
8. B. Kruglikov, V. S. Matveev, “The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta”, Nonlinearity, 29:6 (2016), 1755–1768
9. M. V. Pavlov, S. P. Tsarev, “On local description of two-dimensional geodesic flows with a polynomial first integral”, J. Phys. A, 49:17 (2016), 175201, 20 pp.
10. Yu. Yu. Bagderina, “Ratsionalnye integraly vtoroi stepeni dvumernykh uravnenii geodezicheskikh”, Sib. elektron. matem. izv., 14 (2017), 33–40
11. N. V. Denisova, “Polynomial integrals of mechanical systems on a torus with a singular potential”, Dokl. Phys., 62:8 (2017), 397–399
12. Nikolay A. Kudryashov, “Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada – Kotera and Kupershmidt Equations”, Regul. Chaotic Dyn., 25:1 (2020), 59–77