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Regular and Chaotic Dynamics

, Volume 21, Issue 7–8, pp 821–831 | Cite as

On the extendability of Noether’s integrals for orbifolds of constant negative curvature

  • Valery V. Kozlov
Nonlinear Dynamics & Mobile Robotics

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 

MSC2010 numbers

34A34 58E10 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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