three-body problemA specific case of the n-body problem in which the trajectories of three mutually interacting bodies are considered. There is no general solution for the problem although solutions exist for a few special instances. Thus the orbits can be determined if one of the bodies has negligible mass, as in the case of a planetary satellite, such as the Moon, subject to perturbations by the Sun or an asteroid whose motion is perturbed by Jupiter.
in astronomy, the problem of the motion of three bodies that attract each other in accordance with Newton’s law of gravitation and are regarded as mass points (seeTWO-BODY PROBLEM). The classic example of the three-body problem deals with the system consisting of the sun, the earth, and the moon.
In 1912 the Finnish astronomer K. F. Sundmann found a general solution of the three-body problem in the form of series that converge for any moment of time t. Sundmann’s series, however, turned out to be useless for practical calculations because of their extremely slow convergence.
Under certain special initial conditions it is possible to obtain very simple solutions of the three-body problem; such solutions include those found by Lagrange and are of great interest for astronomy (seeLIBRATION POINTS). A special case of the three-body problem is the elliptical restricted three-body problem, in which two bodies of finite mass move about a center of inertia in elliptical orbits, and the third body has an infinitely small mass. Various classes of periodic motions have been investigated for the restricted problem.
The properties of motion in the limit as t → +∞ and t → +∞, that is, terminal motions, have been studied in detail for the general three-body problem.
G. A. CHEBOTAREV