N-body problem
The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e. Newton's laws of motion and Newton's law of gravity.
The two-body problem is the simplest case: its solution is that each body travels along a conic section which has a focus at the centre of mass of the system. See also Kepler's first law of planetary motion.
The three-body problem is much more complicated; its solution can be chaotic. In general, the three-body problem cannot be solved analytically, although approximate solutions can be calculated by numerical methods or perturbation methods.
The circular restricted three-body problem is the special case in which two of the bodies are in circular orbits and the third is of negligible mass (approximated by the Sun - Earth - Moon system). The restricted problem (both circular and elliptical) was worked on extensively by many famous mathemeticians and physicists, notably Lagrange in the 18th century and Henri Poincaré in at the end of the 19th century. Poincare's work on the restricted three-body problem was the foundation of deterministic chaos. In the circular problem, there exist five equilibrium points. Three are colinear with the masses (in the rotating frame) and are unstable. The remaining two are located 60 degrees ahead of and behind the smaller mass (e.g., Jupiter) in its orbit about the larger mass (e.g., Sun), forming two equilateral triangles. For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.
In 1912, Finland-Swedish mathematician Karl Frithiof Sundman developed a convergent infinite series that provides a solution to the restricted three-body problem. Unfortunately, getting the series to converge to any useful precision requires so many terms (on the order of 108,000,000) that his solution is of little practical use.
See also: many-body problem