billiards. Example 1.2 is quite old, and I do not know
considered for the first time. The next example, altho
the previous one, is surprisingly recent; see [45, 29].
Example 1.6. Consider three elastic point-masses mi,
circle. We expect this mechanical system also to be is
billiard.
Let #i, #2 #3
De
the angular coordinates of the po
ing
S1
as R/27rZ, lift the coordinates to real numbers
lifted coordinates by the same letters with bar (this lift
one may change each coordinate by a multiple of 2TT
coordinates as in Example 1.2. Collisions between p
correspond to three families of parallel planes in thr
space:
+ 27r/c, = + 27rra, =
y/rfii y/rn^ y/rrfi y/rn* y/rn*
where fc, m, n G Z.
All the planes involved are orthogonal to the plan
(1.8) \Jm\X\ + \Jvfi2X2 + y/rn^xs = const,
and they partition this plane into congruent triangle
partition space into congruent infinite triangular prism
tem of three point-masses on the circle is isomorphic
inside such a prism. The dihedral angles of the prism
computed in Exercise 1.4 b).
Arguing as in Exercise 1.4 c), one may reduce one
dom. Namely, the center of mass of the system has the
mivi + m2V2 + m3vs
mi + ra2 + ra3
One may choose the system of reference at this center
in the new coordinates, means that
mivi + y/rri2V2 + y/rn3V3 = 0,
Previous Page Next Page