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
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