"Describing the motion of any planetary system (including purely imaginary ones that exist only on paper) is the subject of a branch of mathematics called celestial mechanics. Its problems are extremely difficult and have eluded the greatest mathematicians in history."
by Paul Trow, Chaos and the Solar System
Lagrange
Joseph-Louis Lagrange showed that there were at least some solutions to the three body problem if we restricted the three bodies to move in the same plane and assumed that the mass of one of them was so small as to be negligible. In his solutions, the three bodies move in unison, always maintaining the same positions relative to each other.
If we think of a two body system (like the simplified Earth and sun system considered above) then the points at which a third small body may be found are now known as the Lagrange points in his honour. It turns out that there are five of these (see diagram below).
