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 Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 206–217 (Mi tm3759)

On the stability of periodic trajectories of a planar Birkhoff billiard

A. P. Markeev

Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526 Russia

Abstract: The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem.

 Funding Agency Grant Number Russian Science Foundation 14-21-00068 This work is supported by the Russian Science Foundation under grant 14-21-00068 and performed at the Moscow Aviation Institute (National Research University).

DOI: https://doi.org/10.1134/S0371968516040129

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 190–201

Bibliographic databases:

UDC: 531.36+531.538

Citation: A. P. Markeev, “On the stability of periodic trajectories of a planar Birkhoff billiard”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 206–217; Proc. Steklov Inst. Math., 295 (2016), 190–201

Citation in format AMSBIB
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