
Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2018, Volume 18, Issue 1, Pages 25–39
(Mi isu742)




Scientific Part
Mathematics
Stability of periodic billiard trajectories in triangle
A. N. Kirillov^{ab}, R. V. Alkin^{a} ^{a} Petrozavodsk State University, 33, Lenin Str., Petrozavodsk, Republic of Karelia,
Russia, 185910
^{b} Institute of Applied Mathematical Research of
the Karelian Research of the Russian Academy of Science, 11, Pushkinskaya Str., Petrozavodsk, Republic
of Karelia, Russia, 185910
Abstract:
The problem of stability of periodic billiard
trajectories in triangles is considered. The notion of stability
means the preservation of a period and qualitative structure of a
trajectory (its combinatorial type) for sufficiently small
variations of a triangle. The geometric, algebraic and fan
unfoldings are introduced for stable trajectories description. The
new method of fan coding, using these unfoldings, is proposed.
This method permits to simplify the stability analysis. The notion
of code equivalence and combinatorial type of a trajectory is
proposed for trajectories classification. The rigorous definition
of stable periodic trajectory in a triangle is formulated. The
necessary and sufficient conditions of a fan code stability are
obtained (Theorem 1). In order to simplify the stable periodic
trajectories classification the notion of pattern, is introduced
which permits us to generate the stable codes (Theorem 2). The
method of stable periodic trajectories construction is proposed
(Theorem 3). The introduced notions are illustrated by several
examples, particularly for trajectories in obtuse triangles. The
possibility of application of the developed instrument to obtuse
triangles offers opportunities of its using to solve the problem
of the existence of periodic billiard trajectories in obtuse
triangles. A new notion of periodic billiard trajectory
conditional stability, relating to some special variations, is
introduced.
Key words:
mathematical billiard, coding of trajectories, stability, pattern, fan code.
DOI:
https://doi.org/10.18500/1816979120181812539
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517.938
Citation:
A. N. Kirillov, R. V. Alkin, “Stability of periodic billiard trajectories in triangle”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 18:1 (2018), 25–39
Citation in format AMSBIB
\Bibitem{KirAlk18}
\by A.~N.~Kirillov, R.~V.~Alkin
\paper Stability of periodic billiard trajectories in triangle
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2018
\vol 18
\issue 1
\pages 2539
\mathnet{http://mi.mathnet.ru/isu742}
\crossref{https://doi.org/10.18500/1816979120181812539}
\elib{http://elibrary.ru/item.asp?id=35647728}
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