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.
Similar content being viewed by others
References
A. M. Abdrakhmanov, “On the stability of two-segment periodic trajectories of a billiard on two-dimensional surfaces of constant curvature,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 4, 88–90 (1991) [Moscow Univ. Mech. Bull. 46(4), 21–23 (1991)].
V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Springer, Berlin, 2006), Encycl. Math. Sci.3.
V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of Diffraction of Short Waves (Nauka, Moscow, 1972); Engl. transl.: Short-Wavelength Diffraction Theory: Asymptotic Methods (Springer, Berlin, 1991).
G. D. Birkhoff, “On the periodic motions of dynamical systems,” Acta Math. 50, 359–379 (1927).
G. D. Birkhoff, Dynamical Systems (Am. Math. Soc., New York, 1927), Colloq. Publ.9.
G. A. Gal’perin and A. N. Zemlyakov, Mathematical Billiards: Billiard Problems and Related Issues of Mathematics and Mechanics (Nauka, Moscow, 1990) [in Russian].
A. P. Ivanov, Dynamics of Systems with Mechanical Impacts (Mezhdunar. Programma Obrazovaniya, Moscow, 1997) [in Russian].
A. P. Ivanov and A. G. Sokol’skii, “On the stability of a nonautonomous hamiltonian system under a parametric resonance of essential type,” Prikl. Mat. Mekh. 44(6), 963–970 (1980) [J. Appl. Math. Mech. 44, 687–692 (1981)].
S. O. Kamphorst and S. Pinto-de-Carvalho, “The first Birkhoff coefficient and the stability of 2-periodic orbits on billiards,” Exp. Math. 14(3), 299–306 (2005).
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge Univ. Press, Cambridge, 1997).
V. V. Kozlov, “A constructive method of establishing the validity of the theory of systems with non-retaining constraints,” Prikl. Mat. Mekh. 52(6), 883–894 (1988) [J. Appl. Math. Mech. 52, 691–699 (1988)].
V. V. Kozlov, “Two-link billiard trajectories: Extremal properties and stability,” Prikl. Mat. Mekh. 64(6), 942–946 (2000) [J. Appl. Math. Mech. 64, 903–907 (2000)].
V. V. Kozlov, “Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard,” Tr. Mat. Inst. im._V.A. Steklova, Ross. Akad. Nauk 273, 212–230 (2011) [Proc. Steklov Inst. Math. 273, 196–213 (2011)].
V. V. Kozlov and I. I. Chigur, “The stability of periodic trajectories of a billiard ball in three dimensions,” Prikl. Mat. Mekh. 55(5), 713–717 (1991) [J. Appl. Math. Mech. 55, 576–580 (1991)].
V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Am. Math. Soc., Providence, RI, 1991), Transl. Math. Monogr.89.
I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966) [in Russian].
A. A. Markeev, “Stability of motion in some problems of the dynamics of systems with non-retaining constraints,” Cand. Sci. (Phys.–Math.) Dissertation (Moscow State Univ., Moscow, 1995).
A. P. Markeev, “Area-preserving mappings and their applications to the dynamics of systems with collisions,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 37–54 (1996) [Mech. Solids 31(2), 32–47 (1996)].
A. P. Markeev, “Stability of equilibrium states of Hamiltonian systems: A method of investigation,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 3–12 (2004) [Mech. Solids 39(6), 1–8 (2004)].
A. P. Markeev, “A method for analytically representing area-preserving mappings,” Prikl. Mat. Mekh. 78(5), 611–624 (2014) [J. Appl. Math. Mech. 78, 435–444 (2014)].
A. P. Markeev, “Simplifying the structure of the third-and fourth-degree forms in the expansion of the Hamiltonian with a linear transformation,” Nelinein. Din. 10(4), 447–464 (2014).
A. P. Markeev, “On the stability of fixed points of area-preserving maps,” Nelinein. Din. 11(3), 503–545 (2015).
A. P. Markeev, “On the stability of a two-link trajectory of a parabolic Birkhoff billiard,” Nelinein. Din. 12(1), 75–90 (2016).
J. K. Moser, Lectures on Hamiltonian Systems (Am. Math. Soc., Providence, RI, 1968), Mem. AMS81.
H. Poincaré, Les méthodes nouvelles de la mécanique céleste (Librairie Scientifique et Technique Albert Blanchard, Paris, 1987), Vol. 3; Engl. transl.: New Methods in Celestial Mechanics (Am. Inst. Phys., Bristol, 1993), Vol.3.
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (Springer, Berlin, 1971).
Ya. G. Sinai, “Dynamical systems with elastic reflections,” Usp. Mat. Nauk 25(2), 141–192 (1970) [Russ. Math. Surv. 25(2), 137–189 (1970)].
S. Tabachnikov, Geometry and Billiards (Am. Math. Soc., Providence, RI, 2005; Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2011).
D. V. Treshchev, “On the question of the stability of the periodic trajectories of Birkhoff’s billiard,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 2, 44–50 (1988) [Moscow Univ. Mech. Bull. 43(2), 28–36 (1988)].
D. Treschev, “Billiard map and rigid rotation,” Physica D 255, 31–34 (2013).
D. V. Treschev, “On a conjugacy problem in billiard dynamics,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 289, 309–317 (2015) [Proc. Steklov Inst. Math. 289, 291–299 (2015)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.P. Markeev, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 206–217.
Rights and permissions
About this article
Cite this article
Markeev, A.P. On the stability of periodic trajectories of a planar Birkhoff billiard. Proc. Steklov Inst. Math. 295, 190–201 (2016). https://doi.org/10.1134/S0081543816080125
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543816080125