THE BILLIARD PROBLEM

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 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 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. Now cut the table along the x axis, and pull the two semicircular halves apart (along y) a distance . Then fill in these two open sections with straight segments. Thus yields a circular table, while nonzero values of give a table with a more traditional stadium shape. The figure shows the trajectory of a billiard moving on a table with . This motion is chaotic.

 
Figure 1: Trajectory of a billiard on a stadium shaped table with .

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 distance between the trajectories of two such similar billiards is a good indicator of the nature of the system. If this distance goes to zero, or stays constant, 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 the billiard problem is that the chaotic behavior is evident even for quite small values of . It turns out that the stadium billiard is chaotic for any nonzero value of . 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 is developed further in our discussion of entropy and the approach to equilibrium in Chapter 7.