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Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 53–71 (Mi tm3747)  

This article is cited in 3 scientific papers (total in 3 papers)

Degenerate billiards

S. V. Bolotin

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

Abstract: In an ordinary billiard system, trajectories of a Hamiltonian system are elastically reflected after a collision with a hypersurface (scatterer). If the scatterer is a submanifold of codimension more than $1$, we say that the billiard is degenerate. We study those trajectories of degenerate billiards that have an infinite number of collisions with the scatterer. Degenerate billiards appear as limits of systems with elastic reflections or as small-mass limits of systems with singularities in celestial mechanics. We prove the existence of trajectories of such systems that shadow the trajectories of the corresponding degenerate billiards. The proofs are based on a version of the method of an anti-integrable limit.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.1134/S037196851604004X

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 45–62

Bibliographic databases:

Document Type: Article
UDC: 531.01
Received: June 22, 2016

Citation: S. V. Bolotin, “Degenerate billiards”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 53–71; Proc. Steklov Inst. Math., 295 (2016), 45–62

Citation in format AMSBIB
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\publ MAIK Nauka/Interperiodica
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  • https://doi.org/10.1134/S037196851604004X
  • http://mi.mathnet.ru/eng/tm/v295/p53

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Sergey V. Bolotin, “Degenerate Billiards in Celestial Mechanics”, Regul. Chaotic Dyn., 22:1 (2017), 27–53  mathnet  crossref  mathscinet
    2. S. V. Bolotin, V. V. Kozlov, “Topological approach to the generalized $n$-centre problem”, Russian Math. Surveys, 72:3 (2017), 451–478  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. J. Fejoz, A. Knauf, R. Montgomery, “Lagrangian relations and linear point billiards”, Nonlinearity, 30:4 (2017), 1326–1355  crossref  mathscinet  zmath  isi  scopus
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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