n-Body Problems

Scientists have always been fascinated by the motion of the planets in the solar system. Around the turn of the century, this fascination led mathematician Henri Poincar\'e to develop a new branch of mathematics, now known as dynamical systems.

In looking at such problems, Poincar\'e and other mathematicians who followed him did not confine themselves to looking only at the solar system. They considered the more general setting in which any number of planets move in space under the influence of their mutual gravitational attraction. This class of problems is called ``n-body'' problems, where n stands for the number of planets (often researchers refer to them as particles rather than planets).

In the solar system, the sun has so much more mass than the planets that the planets' gravitational attraction to the sun dominates all planetary motion. But in a system where the particles involved are closer in mass, the complex patterns of their mutual attractions come into play. Then almost anything can happen: stable orbits can develop, chaotic motions can set in, there can be collisions, or a particle can escape from the system and go flying off to infinity.

The article, '' Off to Infinity in Finite Time '' by Donald G. Saari and Zhihong (Jeff) Xia, published in the May 1995 issue of the Notices of the AMS, explores an intriguing theoretical question that has confounded mathematicians since the time of Poincar\'e: Is it possible that, without any of the particles colliding, one of them could still get up enough acceleration to get infinitely far from all the other particles in a finite amount of time?

The surprising and counterintuitive answer is yes---as long as you have more than four particles. Xia's work that shows that this strange behavior can occur if one has five or more particles. Three particles are not enough; for four particles, Saari's work shows that such behavior is unlikely to occur, though it is not yet known whether it is completely impossible.

-Allyn Jackson