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
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