N-Body Gravitational Problem
The n-body gravitational problem is the mathematical formulation of a question that has fascinated and puzzled astronomers for millennia: How can we predict the motion of the moon and the planets?
According to Isaac Newton's theory of gravitation, any two objects (or "bodies") attract each other by a gravitational force inversely proportional to the square of their distance from each other. Furthermore, Newton's second law of motion states that the acceleration of each body is proportional to the sum of the forces on it from each of the other bodies. Thus, if a certain number (n) of bodies are released at a given time, the acceleration of each one can be computed precisely from the position of the others. Their motion is completely determined, for all time, by their initial positions and velocities.
Newton's theory would appear to be the end of the n-body gravitational problem, but in fact it was just the beginning. One reason is that Newton's formulation was highly idealized: It assumed all the objects are "point masses." (That is, they occupy only a single point in space.) Thus it omitted tidal effects, which can be quite important in the actual solar system. Newton also, of course, ignored the effects of relativity theory, which were discovered by Albert Einstein over 200 years later.
But these are physical objections. There were also pressing mathematical problems, namely that the theory gave a differential equation that the motion of the n bodies must obey, but did not say how to solve that differential equation. The reputation of Newton's theory was cemented by the fact that he did succeed in solving the equation exactly for the case of two bodies: The two bodies travel in ellipses. Philosophically, this means that the orbits are stable for all time. It also means the two bodies can never collide unless they start out on a collision course, which can be considered "infinitely unlikely." (Imagine trying to fire two bullets so that they collide in midair.)
Newton was unable to perform the same feat for three bodies, even the three bodies of most consequence to astronomers: the earth, the moon, and the sun. Even today, only a very few special configurations of three bodies can be solved exactly, most notably a rotating equilateral triangle with the sun at one vertex, the earth at another, and a very small object, such as a satellite, at the third vertex, called a Lagrange point. Such a configuration has actually been used for some space missions.
Lacking exact solutions, mathematicians have made much more progress on the qualitative, or philosophical, questions related to the n-body problem: Is the motion stable? Is it predictable, in the sense that small errors in observation of a planet will only lead to small errors in the prediction of its motion? How often do collisions occur? Are other "singularities" possible, such as the ejection of a planet to an infinite distance away in a finite amount of time?
At the risk of oversimplifying a beautiful and deep theory, here are the major results on these questions, along with the names of the mathematicians who proved them. A very small perturbation of an exactly solvable system (such as the perturbation of one planet's motion by another) is likely--but not certain--to be stable for all time. (Kolmogorov, Arnold, Moser) Collisions continue to be "infinitely unlikely" (Saari); however, bear in mind here that we are talking about systems of point masses, since collisions do occur in the real solar system. Ejection singularities are not possible for the three-body problem, but are possible for five or more bodies (Painlevé, Xia, Gerver). Perhaps most importantly, the solutions to the n-body problem are usually not predictable over the long term but are chaotic (Poincaré, Birkhoff, Smale). The last theorem, in particular, explains why it is futile to hope for an exact general solution of the three (or more)-body problem. We can only settle for short-term approximations by computer, such as those NASA uses when planning space missions, or long-term solutions in special cases, such as the Lagrange configuration.
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