I am interested to learn to what extent results on billiards
in polygons have been extended to multiple balls.
Assume the balls have equal radii and the same mass,
the same initial speed, and all
collisions are perfectly elastic, as depicted in this* image:
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
![Wikipedia image][1]
<br />
To be specific, let me start with just the square table.
For single particle billiards, it is well known
that (1) a trajectory of rational slope that avoids the corners
is periodic, and (2) a trajectory of irrational slope that avoids
the corners will be "uniformly distributed" in the sense
that it spends equal times in equal areas.

> Are there analogous results for billiard systems of $n>1$ balls
within a square?

Perhaps it is necessary to make some assumption concerning
the size of the ball radii and the box dimensions?
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
![3 balls][2]
<br />
The literature I've seen on billiard dynamics does not
explore this territory.  Likely there are results in the literature,
in which case pointers would be appreciated.  Thanks!
<hr />
*This impressive [Wikipedia image][3] by A.Greg was once the "Picture of the Day."


  [1]: http://upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif
  [2]: http://cs.smith.edu/~orourke/MathOverflow/ThreeBalls.jpg
  [3]: http://en.wikipedia.org/wiki/Elastic_collision