Dynamical billiards
From Freepedia
A billiard is a dynamical system where a particle alternates between motion in a straight line and specular reflections with a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be many dimensional. The outer counterpart of billiards is known as outer billiard theory.
The motion of the particle in the billiard is a straight line between reflections with the boundary (a geodesic if the table has any curvature). All reflections are specular: the angle of incidence just before the collision is equal to the angle of reflection just after the collision. The sequence of reflections is called the billiard map and completely characterizes the motion of the particle.
Billiards capture all the complexity of Hamiltonian systems, from integrability to chaotic motion, without the difficulties of integrating the equations of motion to determine its Poincaré map. Birkhoff showed that a billiard system with an elliptic table is integrable. The two well known chaotic billiards are the Sinai billiard and the Bunimovich stadium.
Contents |
Notable billiard tables
Sinai billiard
The table of the Sinai billiard is a square with a disk removed from its center. The billiard arises from studying the behavior of two interacting disks bouncing inside a square, reflecting off the boundaries of the square and off each other. By eliminating the center of mass as a configuration variable, the dynamics of two interacting disks reduces to the dynamics in the Sinai billiard.
The billiard was introduced by Sinai as an example of an interacting Hamiltonian system that displays physical thermodynamic properties: it is ergodic and has a positive Lyapunov exponent. As a model of a classical gas, the Sinai billiard is sometimes called the Lorentz gas.
Bunimovich stadium
The table called the Bunimovich stadium is a rectangle capped by semi-circles. Until it was introduced by Leonid Bunimovich, billiards with positive Lyapunov exponents where thought to need convex scatters, such as the disk in the Sinai billiard, to produce the exponential divergence of orbits. Bunimovich showed that by considering the orbits beyond the focusing point of a concave region it was possible to obtain exponential divergence.
Quantum chaos
Billiards are well suited for semiclassical calculations, as their periodic orbits can be easily computed. Both stadium billiards and Sinai billiards are routinely created in physics laboratories (for example, by using atom optics) to study classical and quantum chaos.
References
- Ya. Sinai, Dynamical systems with elastic reflections, Russian Math. Surveys, 25, p.137 (1970)
- V. I. Arnold and A. Avez, Théorie ergodique des systèms dynamiques, (1967), Gathier-Villars, Paris. (English edition: Benjamin-Cummings, Reading, Mass. 1968). (Provides discussion and references for Sinai's billiards.)
- S. Sridhar and W. T. Lu, Sinai Billiards, Ruelle Zeta-functions and Ruelle Resonances: Microwave Experiments, (2002) Journal of Statistical Physics, Vol. 108 Nos. 5/6, pp. 755-766.
- Linas Vepstas, Sinai's Billiards, (2001). (Provides ray-traced images of Sinai's billiards in three-dimensional space. These images provide a graphic, intuitive demonstration of the strong ergodicity of the system.)
