of radius r on the "unit" torus R 2 /Z 2 . The positio

characterized by its center, a point on the torus. If x\

positions of the two centers, then the distance betwee

not less than 2r. The set of such pairs (#1,22) is the

space of our system. Each X{ can be lifted to R2; such

up to addition of an integer vector. However, the veloc

defined vector in R2.

Figure 1.5. Reduced configuration space of two discs on

Similarly to Example 1.6, one can reduce the num

of freedom by fixing the center of mass of the system

that we consider the difference x = X2 — x\ which is

torus at distance at least 2r from the point representin

R2; see figure 1.5. Thus the reduced configuration spa

with a hole, a disc of radius 2r. The velocity of this

point is the vector i2 — v\.

When the two discs collide, the configuration p

boundary of the hole. Let v be the velocity of poin

collision and u after it. Then we have decompositions

The law of reflection implies that the tangential com

change: u\ = v\,u\ —

vl2.

To find u\ and u^, use (1.1)

The solution of this system is: u[ = v^u^ — v\. He

v\)

~

(v2

~~

vi)-

Note that the vector v\ — v\ is perpen

thus tangent to the boundary of the configuration sp

vector

vr2

— v\ is collinear with x and hence normal to