11 - Chaos

Summary

hannah: The weather is fairly predictable in the Sahara

valentine: The scale is different but the graph goes up and down the same way. Six thousand years in the Sahara looks like six months in Manchester, I bet you.

from Arcadia by Tom Stoppard

It is time to correct and soften a striking difference in emphasis between the chapters on Mechanics (6) and Statistical Mechanics (8). Contrast the simple and regular motions in the former with the unpredictable jigglings in the latter. The message here will be that even in mechanics easy predictability is not by any means universal, and generally found only in carefully chosen simple examples. Sensitivity to precise ‘aiming’ is found in problems just slightly more complicated than those we considered in our discussion of mechanics. However, the time it takes for such sensitivity to manifest itself in perceptibly different trajectories depends on the particular situation. ‘Chaotic’ as a technical term has come to refer to trajectories with a sensitive dependence on initial conditions, and the general subject of their study is now called Chaos.

There is a long tradition of teaching mechanics via simple examples, such as the approximately circular motion of the moon under the gravitational influence of the earth. This is a special case of the ‘two body problem’ – two massive objects orbiting about each other – whose solution in terms of elliptical trajectories is one of the triumphs of Newton's Laws.