2.4 Poincaré’s error
In the restricted three-body problem, two of the bodies move as a two-body problem and so their motion is known. The third body (often called the planetoid) is considered massless relative to the other two larger bodies – you can think of it as a speck of dust. The motion of the planetoid does not affect the motion of the two larger bodies, but its motion is affected by them. The problem is then to ascertain the orbit of the small body.
This might seem a very artificial problem, but in fact it provides a good model for the Sun-Earth-Moon system (in which the Moon is the planetoid). But Poincaré was unable to solve even this version of the problem completely. Nevertheless, he won the prize because he developed a lot of new and important mathematical techniques in his efforts to do so. But his path to glory was not smooth. In fact, it turned out that the first paper he submitted to the competition – the paper for which he won the prize – had a serious error. This was discovered while the paper was being prepared for publication. Fortunately, Poincaré was able to correct the error before the prize ceremony. Unfortunately, he did have to pay for the reprinting of his memoir, which cost him more than he won in prize money!
But Poincaré’s error had further-reaching consequences than he could possibly have imagined when he corrected the paper. Originally, Poincaré had assumed (without justification) that the orbit of the small body was stable. What does this mean? Suppose that, given the small body’s initial position and velocity, its orbit can be predicted at any time in the future (or in the past). Poincaré’s assumption from here was that given very slightly different initial conditions – tiny changes to the small body’s position and velocity – its orbit will remain close to before.
But, as Poincaré himself discovered, this assumption was a mistake. The behaviour of the small body, although governed by deterministic laws, was generally unpredictable, and as Poincaré himself said:
It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible.
In the following video, which simulates the restricted three-body problem, the two large bodies are moving in a rotating reference frame so that they both appear stationary. The small body is in fact five small bodies starting extremely close together. In Poincaré’s words, these five bodies exhibit ‘very small differences in the [five sets of] initial conditions’. After three minutes the five bodies diverge, demonstrating the system’s sensitive dependence on initial conditions.
