one remains the same. In particular, the speed of the
change, and one may assume that the point always m
This description of the billiard reflection applies
multi-dimensional space and, more generally, to other g
only to the Euclidean one. Of course, we assume that
occurs at a smooth point of the boundary. For example
ball hits a corner of the billiard table, the reflection is n
the motion of the ball terminates right there.
There are many questions one asks about the b
many of them will be discussed in detail in these notes
let D be a plane billiard table with a smooth boun
interested in 2-periodic, back and forth, billiard traject
In other words, a 2-periodic billiard orbit is a segme
D which is perpendicular to the boundary at both en
following exercise is rather hard; the reader will have
Chapter 6 for a relevant discussion.
Exercise 1.1. a) Does there exist a domain D witho
b) Assume that D is also convex. Show that there exi
distinct 2-periodic billiard orbits in D.
c) Let D b e a convex domain with smooth boundary in t
space. Find the least number of 2-periodic billiard orb
d) A disc D in the plane contains a one parameter famil
billiard trajectories making a complete turn inside D
tories are the diameters of D). Are there other plane
tables with this property?
In this chapter, we discuss two motivations for the
ematical billiards: from classical mechanics of elastic
from geometrical optics.
Example 1.2. Consider the mechanical system con
point-masses rai and rri2 on the positive half-line x