Outer billiard
From Wikipedia, the free encyclopedia
Outer billiard is a dynamical system that differs from a usual dynamical billiard in that it deals with a discrete sequence of moves in the outer region, instead of the continuous motion of a mass-point (or billiard ball) inside of a "billiard table", a bounded plane region with a piecewise smooth boundary endowed with a Riemannian metric.
The outer (or dual) billiard transformation is defined as follows: let P be a billiard table and x a ball positioned in the exterior of P. Then there are two points on P that form support lines through x. The outer billiard transformation F, which comes with a clockwise or counterclockwise orientation, maps x to its reflection through the support point in the given direction. An orbit is the set of all iterations of a point around a table, and a point (and its orbit) are called periodic if the point maps back onto itself.
Outer billiards were introduced in 1945 by M. Day and attracted attention from Jürgen Moser and others in relation with the stability problem in celestial mechanics. They have been studied in the Euclidean plane as well as the hyperbolic plane and higher dimensions. Bernhard Neumann posed the question of whether or not orbits of the outer billiard map can escape to infinity. This question has been answered in the hyperbolic plane for certain tables, classified as “large”, all of whose orbits escape to infinity. Rich Schwartz has shown that the outer billiard about the Penrose kite in the Euclidean plane has unbounded orbits. Many problems about outer billiards remain open.
[edit] References
- F. Dogru, S. Tabachnikov (2003). "On Polygonal Dual Billiards in the Hyperbolic Plane". Regular Chaotic Dynamics 8: 67–82. doi:.
- J. Moser (1978). "Is the Solar System Stable?". Mathematical Intelligencer 1: 65–71.
- R.E. Schwartz "Unbounded Orbits for Outer Billiards" Penn State 2007
- S. Tabachnikov (1995). Billiards. SMF Panoramas et Syntheses. ISBN 2-85629-030-2.
- S. Tabachnikov (2002). "Dual Billiards in the Hyperbolic Plane". Nonlinearity 15: 1051–1072. doi:.
