N-Body Problem

Everywhere we look in the Universe, we see clusters of objects. Most stars are not single, like the Sun; they consist of two or more stars orbiting one another. Groups of stars are found in the open and globular star clusters, the latter containing up to a million stars. Our Milky Way galaxy is a huge swarm of a few hundred billion stars, and galaxies themselves are grouped into clusters.

These clusters of objects, whether they be individual stars, star clusters, clouds of gas, or groups of galaxies, are impelled to move relative to one another in accordance with the law of gravitation. In most circumstances, the law of gravity as developed by Isaac Newton describes the motion of these objects adequately. (In some extreme conditions, such as around very dense objects with intense gravitational fields, and even in very high-precision analyses of the orbits of more mundane objects like the planet Mercury, Albert Einstein's treatment of gravitation in his theory of general relativity is required.) Newton discovered that the motions of orbiting objects could be explained (and predicted) if the gravitational attraction between two bodies depended on the product of their masses and the inverse square of their distance. The complete, analytical solution of the so-called two-body problem is one of the great triumphs of classical mechanics.

Difficulties arise, however, when one considers more than two bodies. The motions in a system of three interacting bodies, defining (sensibly enough) the three-body problem, cannot fully be treated analytically, meaning that one cannot derive on paper a single equation to predict the positions and velocities of the three bodies in the system at some arbitrary time in the future. An analytic solution is likewise impossible for a larger system of some number N of objects, and this seemingly intractable situation is called the N-body problem.

The N-body problem has many very important applications in astronomy. Wherever the long-term motions of a large group of gravitationally interacting bodies are concerned, the N-body problem looms large. This covers such diverse areas as the evolution of our solar system, the formation and gradual dissipation of clusters of stars, the evolution of galaxies, and even the overall evolution of the clusters of galaxies that comprise the large-scale structure of the Universe. Some approach to solving the N-body problem is clearly desirable.