Three-body problem

In physics and classical mechanics, the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with Newton's laws of motion and of universal gravitation which are the laws of classical mechanics. The three-body problem is a special case of the n-body problem.

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun. In an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles.

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his "Principia" (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the "Principia", and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.

The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499 Vespucci used knowledge of the position of the moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the Marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.

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