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
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).
Lobachevsky plane, Fuchsian group, orbifold, Noether integrals
|Russian Science Foundation
|This work was supported by the grant of the Russian Science Foundation (project No 14–50–00005).
MSC: 34A34, 58E10
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
\by Valery V. Kozlov
\paper On the Extendability of Noether’s Integrals for Orbifolds of Constant Negative Curvature
\jour Regul. Chaotic Dyn.
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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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