Cite this article as:

Kirillov A. N., Alkin R. V. Stability of Periodic Billiard Trajectories in Triangle. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 1, pp. 25-39. DOI: https://doi.org/10.18500/1816-9791-2018-18-1-25-39


Published online: 
28.03.2019
Language: 
Russian
Heading: 
UDC: 
517.938

Stability of Periodic Billiard Trajectories in Triangle

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.

References

1. Cornfeld I. P., Fomin S. V., Sinai Y. G. Ergodic Theory. New York, Springer-Verlag, 1982. 491 p. DOI: https://doi.org/10.1007/978-1-4615-6927-5 (Russ. ed.: Moscow, Nauka. 384 p.)
2. Rademacher H., Toeplitz O. Von Zahlen und Figuren. Berlin, Springer-Verlag, 1933. 173 p. (Russ. ed.: Moscow, Fizmatgiz, 1962. 263 p.)
3. Rubinstein A. I., Telyakovskii D. S. Zamechania o zadache Faniano Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 4, pt. 1, pp. 382–387 (in Russian).
4. Vorobets Ya. B., Gal’perin G. A., Stepin A. M. Periodic billiard trajectories in polygons : generating mechanisms Russian Math. Surveys, 1992, vol. 47, no. 3, pp. 5–80. DOI: https://doi.org/10.1070/RM1992v047n03ABEH000893
5. Schwartz R. E. Obtuse Triangular Billiards II: One Hundred Degrees Worth of Periodic Trajectories Experimental Math., 2009, vol. 18, iss. 2, pp. 137–171. DOI: https://doi.org/10.1080/10586458.2009.10128891
6. Kozlov V. V. Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard. Proc. Steklov Inst. Math., 2011, vol. 273, pp. 196–213. DOI: https://doi.org/10.1134/S0081543811040092
7. Markeev A. P. On the stability of the two-link trajectory of the parabolic Birkhoff billiards Rus. J. Nonlin. Dyn., 2016, vol. 12, no 1, pp. 75–90. DOI: https://doi.org/10.20537/nd1601005
8. Kravsov V. M., Kalakova G. K. Geometrija billiardnih traektorij v mnogougolnikah [Geometry of billiard trajectories in polygons]. Saint-Petersburg, EVRASIA, 2013. 304 p. (in Russian).

Short text (in English): 
Full text: