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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 7-8, Pages 821–831 (Mi rcd228)

This article is cited in 1 scientific paper (total in 1 paper)

On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature

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 the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).

Keywords: Lobachevsky plane, Fuchsian group, orbifold, Noether integrals

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work was supported by the grant of the Russian Science Foundation (project No 14–50–00005).

DOI: https://doi.org/10.1134/S1560354716070054

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MSC: 34A34, 58E10
Received: 26.10.2016
Accepted:30.11.2016
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Citation: 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

Citation in format AMSBIB
\Bibitem{Koz16} \by Valery V. Kozlov \paper On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 7-8 \pages 821--831 \mathnet{http://mi.mathnet.ru/rcd228} \crossref{https://doi.org/10.1134/S1560354716070054} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3626354} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000403091800005} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85016016185} 

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This publication is cited in the following articles:
1. V. V. Kozlov, “On reversibility in systems with a non-compact configuration space and non-negative potential energy”, Pmm-J. Appl. Math. Mech., 81:4 (2017), 250–255
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