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{\bf THE BILLIARD PROBLEM}
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The game of billiards is closely connected to a very simple, but
extremely interesting, chaotic system.
Consider a ball
moving without friction on a horizontal table.  We imagine that
there are perfectly reflecting walls at the 
edges of the table, and that there
is no frictional force between the ball and the table.  One can think
of this as a billiard ball that moves without friction on a ``perfect''
billiard table (for simplicity we ignore any effects due to the spin
of the ball). 
The ball is given some initial velocity and the problem
is to calculate and understand the resulting trajectory.  
This is known as the {\em stadium billiard} problem.

Except for the collisions with the walls, the motion of the billiard 
is quite simple.  Between collisions the velocity is constant, and
the behavior can be obtained with a simple
Euler algorithm.  Since the velocity is constant
(except during the collisions),
the Euler solution turns out to give an exact description of
the motion across the table.
The most difficult part
of the calculation is the treatment of the collisions.
Since we assume that they are perfectly elastic,
the reflections will be mirror-like, which means that the angle
of incidence will be equal to the angle of reflection.
The treatment of these collisions requires a little vector
analysis, and is discussed in the book.

When one examines the trajectory of a billiard, it turns out that
the behavior depends on the {\em shape} of the table.
The behavior for a square table is
quite ``regular''; it has a very
simple, predictable, pattern, and is not chaotic.
Things become interesting when one considers
other table shapes.  There are many possibilities; here we will consider
only one, the so-called stadium shape, which can be described
as follows.  Imagine a circular table of radius $r = 1$.
Now cut the table along the $x$ axis, and pull
the two semicircular halves apart (along $y$) 
a distance $2\alpha r$.  Then fill
in these two open sections with straight segments.  Thus $\alpha=0$
yields a circular table, while nonzero values of $\alpha$ give a
table with a more traditional stadium shape.  The figure
shows the trajectory of a billiard moving on
a table with $\alpha = 0.01$.  This motion is chaotic.

\begin{figure}
\centerline{\psfig{figure=stad_0.01_traj.eps,width=3.5in}}
\caption{
{\em
Trajectory of a billiard on a stadium shaped table with $\alpha = 0.01$.  
}
}
\end{figure}

One hallmark of chaotic systems is an extreme
sensitivity to initial conditions.  This property is also found 
in the billiard problem, and can be illustrated nicely if one calculates the
trajectories of two billiards with slightly different initial
positions.  
The {\em distance} between the trajectories of two such similar
billiards is then a good indicator of the nature of the system.  If this
distance goes to zero with time, the motion is regular (i.e., nonchaotic),
while if it grows with time, the motion is chaotic.
The divergence of such trajectories can be
described by a Lyapunov exponent, which is positive in the chaotic regime,
and negative if the motion is nonchaotic.

A remarkable
feature of our results for the billiard problem
is that the chaotic behavior is
evident even for quite small values of $\alpha$.  It turns out that
the stadium billiard is chaotic for {\em any} nonzero value of $\alpha$.
In fact, only tables with very high symmetry are nonchaotic.  
The billiard
problem may be relevant for describing the motion of gas molecules
in a container.  Our results suggest that for any
realistically shaped container (i.e., any shape which
is not extremely symmetric like a
perfectly circular table) such motion is likely to
be chaotic and thus unpredictable.  This finding developed further in
our discussion of entropy and the approach to equilibrium in Chapter~7.

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