Introduction

We saw earlier, in Section 1.9, that an isolated dynamical system consisting of two freely moving point masses exerting forces on one another—which is usually referred to as a two-body problem—can always be converted into an equivalent one-body problem. In particular, this implies that we can exactly solve a dynamical system containing two gravitationally interacting point masses, as the equivalent one-body problem is exactly soluble. (See Sections 1.9 and 3.16.) What about a system containing three gravitationally interacting point masses? Despite hundreds of years of research, no useful general solution of this famous problem—which is usually called the three-body problem—has ever been found. It is, however, possible to make some progress by severely restricting the problem's scope.

Circular restricted three-body problem

Consider an isolated dynamical system consisting of three gravitationally interacting point masses, m1, m2, and m3. Suppose, however, that the third mass, m3, is so much smaller than the other two that it has a negligible effect on their motion. Suppose, further, that the first two masses, m1 and m2, execute circular orbits about their common center of mass. In the following, we shall examine this simplified problem, which is usually referred to as the circular restricted three-body problem. The problem under investigation has obvious applications to the solar system. For instance, the first two masses might represent the Sun and a planet (recall that a given planet and the Sun do indeed execute almost circular orbits about their common center of mass), whereas the third mass might represent an asteroid or a comet (asteroids and comets do indeed have much smaller masses than the Sun or any of the planets).