Volume 8, Number 1

Volume 8, Number 1, 2003
Dynamics of billiards

Gutkin E.
This is a brief survey of the subject, emphasizing the open problems.
Citation: Gutkin E., Billiard dynamics: a survey with the emphasis on open problems on billiards, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 1-13
Bunimovich L. A.
We show that absolute focusing is a necessary condition for a focusing component to be a part of the boundary of a hyperbolic billiard. A sketch of the proof of a general theorem on hyperbolicity and ergodicity of two-dimensional billiards with all three (focusing, dispersing and neutral) components of the boundary is given. The example of a simply connected domain (container) is given, where a system of $N$ elastically colliding balls is ergodic for any $1 \leqslant N <\infty$.
Citation: Bunimovich L. A., Absolute Focusing and Ergodicity of Billiards, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 15-28
Galperin G. A.,  Zvonkine D.
This paper is dedicated to periodic billiard trajectories in right triangles and right-angled tetrahedra. We construct a specific type of periodic trajectories and show that the trajectories of this type fill the right triangle entirely. Then we establish the instability of all known types of periodic trajectories in right triangles. Finally, some of these results are generalized to the $n$-dimensional case and are given a mechanical interpretation.
Citation: Galperin G. A.,  Zvonkine D., Periodic Billiard Trajectories in Right Triangle, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 29-44
Galperin G. A.
It is proved that, in contrast to the smooth convex surfaces homeomorphic to the two-dimensional sphere, the most of convex polyhedra in three dimensional space are free of closed geodesics without self-intersections.
Citation: Galperin G. A., Convex Polyhedra without Simple Closed Geodesics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 45-58
Itin A. P.,  Neishtadt A. I.
We consider an elliptic billiard whose shape slowly changes. During slow evolution of the billiard certain resonance conditions can be fulfilled. We study the phenomena of capture into a resonance and scattering on a resonance which lead to the destruction of the adiabatic invariance in the system.
Citation: Itin A. P.,  Neishtadt A. I., Resonant Phenomena in Slowly Perturbed Elliptic Billiards, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 59-66
Dogru F.,  Tabachnikov S.
We study the polygonal dual billiard map in the hyperbolic plane. We show that for a class of convex polygons called large all orbits of the dual billiard map escape to infinity. We also analyse the dynamics of the dual billiard map when the dual billiard table is a regular polygon with all right angles.
Citation: Dogru F.,  Tabachnikov S., On Polygonal Dual Billiard in the Hyperbolic Plane, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 67-81
Bishop R.
Some of the major concepts of Riemannian geometry are explained in terms of billiards on a circular billiard table: conjugate loci, exponential map, Morse theory on the path space. The conjugate loci are related to the caustics of classical optics of a circular reflector. The change in form of those conjugate loci and caustics as the source point moves is classified and illustrated with many pictures based on numerical data.
Citation: Bishop R., Circular Billiard Tables, Conjugate Loci, and a Cardioid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 83-95
Bedaride N.
We give a new proof for the directional billiard complexity in the cube, which was conjectured in [8] and proved in [10]. Our technique allows us to obtained a similar theorem for some rational polyhedra.
Citation: Bedaride N., Billiard Complexity in Rational Polyhedra, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 97-104
Dragović V.,  Gajić B.
We present the classical Wagner construction from 1935 of the curvature tensor for the completely nonholonomic manifolds in both invariant and coordinate way. The starting point is the Shouten curvature tensor for the nonholonomic connection introduced by Vranceanu and Shouten. We illustrate the construction by two mechanical examples: the case of a homogeneous disc rolling without sliding on a horizontal plane and the case of a homogeneous ball rolling without sliding on a fixed sphere. In the second case we study the conditions imposed on the ratio of diameters of the ball and the sphere to obtain a flat space — with the Wagner curvature tensor equal to zero.
Citation: Dragović V.,  Gajić B., The Wagner Curvature Tenzor in Nonholonomic Mechanics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 105-123
Jovanović B.
In this paper we study the dynamics of the constrained $n$-dimensional rigid body (the Suslov problem). We give a review of known integrable cases in three dimensions and present their higher dimensional generalizations.
Citation: Jovanović B., Some Multidimensional Integrable Cases Of Nonholonomic Rigid Body Dynamics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 125-132

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