David Peat

From Clockwork to Chaos 121Toward the end of the nineteenth century Henri Poincaré attemptedto lay this problem to rest. While he was still unable to solvethe three-body problem specifically, he found a way to say somethinggeneral about the overall shape and behavior of its solutions.Poincaré showed that, in most cases, things turn out as everyoneexpected—small influences result in small effects, and the more accuratesolutions are close to the simpler two-body solutions. Nevertheless,this need not always be the case. In some very exceptional instancesthe perturbed solutions “blow up.” The addition of a large number ofvery small effects accumulates rapidly, and instead of planets being“regular as clockwork,” for certain critical arrangements the systembecomes unstable. In other words, Poincaré had discovered chaos hiddenwithin the heart of the Newtonian universe. Newtonian clockworkwas only regular under certain conditions. Outside this boundary,physics was faced with uncertainty.How does this happen? The rotation of earth round the sun presentsa relatively simple problem for Newtonian science. The sun pullsthe earth toward it; likewise the earth exerts a pull on the sun. Now addin the effect of the moon. As the moon rotates around the earth itexerts a slight pull, which has the effect of slowing down and speedingup the earth’s motion. It is alternately pushing the earth toward thesun and pulling it away again.In the case of the earth–sun system the moon’s effect is not verylarge. But for certain critical arrangements of other planets, “resonance”can take place. To understand resonance, think of a heavy manon a swing and a small child who gives the swing a nudge from time totime. (These nudges can be thought of as perturbations to the motionof the swing.) Over all, these nudges have little effect on the man’sregular swinging back and forth. But suppose the child nudges the maneach time he reaches the highest point of his swing. If each tiny nudgeis timed exactly the nudges will begin to add up. In swing after swingthe man goes higher and higher. This effect of a very small perturbationthat accumulates from oscillation to oscillation is called resonance.In the case of two planets in orbit around the sun the second maybe nudging the first in such a way that these nudges resonate exactlywith that planet’s “year.” In turn, the first planet also nudges the sec-