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
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