conserved. The reflection off the left end point of the h

elastic: if a point hits the "wall" x = 0, its velocity ch

Let x\ and x2 be the coordinates of the points. Th

the system is described by a point in the plane (#i, x2)

inequalities 0 x\ #2- Thus the configuration space

is a plane wedge with the angle 7r/4.

Let vi and v2 be the speeds of the points. As long

do not collide, the phase point {x\,xi) moves with c

(^i

5

^2)- Consider the instance of collision, and let u\, u

after the collision. The conservation of momentum an

as follows:

m\u\

Vfl2u\

m

(l.l) m1ul-\-m2u2 = miVi+m2v2l — h

Introduce new variables: 1,2. In t

the configuration space is the wedge whose lower bo

line x\jy/m{ = x2/y/m2; the angle measure of this we

arctan yjmi/m2 (see figure 1.2).

Figure 1.2. Configuration space of two point-masses

half-line

In the new coordinate system, the speeds rescale

as the coordinates: V\ = yjm{v\, etc. Rewriting (1.1)

„-i2 (1.2) yfm{ U\ + y ^ 2 u2 =

A/^ T

vl + \/^2 v2,

The second of these equations means that the magnitud

ity vector (^1,^2) does not change in the collision. The

in (1.2) means that the dot product of the velocity v