The three-body problem is the mathematical problem of finding the positions and velocities of three massive bodies, which are interacting with each other gravitationally, at any point in the future or the past, given their present positions, masses, and velocities. An example would be to completely solve the behavior of the Sun-Jupiter-Saturn system, or that of three mutually orbiting stars. It is a vastly more difficult exercise than the two-body problem. In fact, as Henri Poincaré and others showed, the three-body problem is impossible to solve in the general case; that is, given three bodies in a random configuration, the resulting motion nearly always turns out to be chaotic: no one can predict precisely what paths those bodies would follow. However, the problem becomes tractable in certain special cases.
In the restricted three-body problem, one of the masses is taken to be negligibly small so that the problem simplifies to finding the behavior of the massless body in the combined gravitational field of the other two. In the circular restricted three-body problem and the elliptical restricted three-body problem, the two masses pursue circular and elliptical orbits, respectively, about their common center of mass. In the coplanar restricted three-body problem the massless body moves entirely in the plane of the massive bodies' orbits; in the three-dimensional three-body problem, it is free to move in all three dimensions. These restricted cases cover systems such as Sun-planet-asteroid, Sun-planet-comet, or binary star-planet.
For three interacting bodies, mathematicians have found a small number of special cases in which the orbits of the three masses are periodic. In 1765, Leonhard Euler discovered an example in which three masses start in a line and rotate so that they stay lined-up; such a set of orbits is unstable, however, and it would be unlikely to occur in nature. Then, in 1772, Joseph Lagrange identified a stable periodic orbit in which three masses, one of which is negligible, at the corners of an equilateral triangle (see Lagrangian points). Each mass moves in an ellipse in such a way that the triangle formed by the three masses always remains equilateral. A Trojan asteroid, which forms a triangle with Jupiter and the Sun, moves according to such a scheme. In 2001, mathematicians Richard Montgomery of the University of California, Santa Cruz and Alain Chenciner of the University of Paris added another exact solution to the equations of motion for three gravitationally interacting bodies. The three equal masses chase each other around the same figure-eight curve in the plane. Computer simulations by Carlès Simò of the University of Barcelona demonstrated that the figure-eight orbit is stable: the orbit persists even when the three masses aren't precisely the same, and it can survive a tiny disturbance without serious disruption. This means there's a chance that the figure-eight orbit might actually be seen in some stellar system. However, it's a pretty small chance-somewhere between one per galaxy and one per universe! The existence of the three-body, figure-eight orbit has prompted mathematicians to look for similar orbits involving four or more masses. Joseph Gerver of Rutgers University, for instance, found one set in which four bodies stay at the corners of a parallelogram at every instant, while each body follows a curve that looks like a figure-eight with an extra twist. Using computers, Simò has found hundreds of exact solutions for the case of N equal masses traveling a fixed planar curve. However, they are not stable and thus of no practical significance.