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,