normal component instantaneously changes sign, while

one remains the same. In particular, the speed of the

change, and one may assume that the point always m

unit speed.

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

billiard trajectory?

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

one remains the same. In particular, the speed of the

change, and one may assume that the point always m

unit speed.

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

billiard trajectory?

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