The Three Body Problem

Poincare etc.

There are two chaotic systems which affect us greatly. The first is the weather. Although weather equations are pretty well understood and are solved by computers every day, it is impossible to take into account all the factors influencing the weather (remember the butterfly). No set of data is perfect, nor are computers perfect at solving the equations. The effects of these small errors build up remarkably quickly. After about ten days it is essentially impossible to forecast weather with any degree of accuracy.

Chaos is also key to understanding the solar system. Whilst the motion of the planets is very predictable, the motion of many of the asteroids is not. Although asteroids do obey Newton's laws, they may well have orbits which move erratically about space. Such an erratic asteroid is thought to have hit the Earth 65 million years ago and wiped out the dinosaurs. The consequences of a similar incident occurring today are unthinkable; the nature of chaotic motion means that such events are virtually impossible to predict until too late.

In 1887, the French mathematician Henri Poincaré showed that while Newton’s theory of gravity could perfectly predict how two planetary bodies would orbit under their mutual attraction, adding a third body to the mix rendered the equations unsolvable. The best we can do for three bodies is to predict their movements moment by moment, and feed those predictions back into our equations … Though the dance of the planets has a lengthy prediction horizon, the effects of chaos cannot be ignored, for the intricate interplay of gravitation tugs among the planets has a large influence on the trajectories of the asteroids. Keeping an eye on the asteroids is difficult but worthwhile, since such chaotic effects may one day fling an unwelcome surprise our way. On the flip side, they can also divert external surprises such as steering comets away from a potential collision with Earth.