the boundary.

We conclude that the (reduced) system of two id

discs on the torus is isomorphic to the billiard on th

disc removed. This billiard system is known as the Sinai

101]. This was the first example of a billiard system t

chaotic behavior; we will talk about such billiards in C

Examples 1.2, 1.6 and 1.10 confirm a general pri

servative mechanical system with elastic collisions is is

certain billiard.

1.2. Digression. Configuration spaces. Introduct

ration space is a conceptually important and non-triv

study of complex systems. The following instructive ex

mon in the Russian mathematical folklore; it is due to

nov (cf. [4]).

Consider the next problem. Towns A and B are

two roads. Suppose that two cars, connected by a

2r, can go from A to B without breaking the rope. P

circular wagons of radius r moving along these roads i

directions will necessarily collide.

To solve the problem, parameterize each road fro

the unit segment. Then the configuration space of p

one on each road, is the unit square. The motion of

A to B is represented by a continuous curve connect

(0,0) and (1,1). The motion of the wagons is represen

connecting the points (0,1) and (1,0). These curves

and an intersection point corresponds to collision of t

figure 1.6.

An interesting class of configuration spaces is pro

linkages, systems of rigid rods with hinge connections

a pendulum is one rod, fixed at its end point; its confi

is the circle S1. A double pendulum consists of two rod

end point; its configuration space is the torus T2 — Sl