Three-Body Problem

The three-body problem is the n-body problem for three bodies under mutual Newtonian gravitational attraction. Unlike the two-body problem, its general motion does not reduce to a single conic section. Nevertheless, special exact solutions and rigorously established periodic motions are known (Heggie 2006).

Euler's collinear solutions and Lagrange's equilateral solutions are generated by central configurations. In the latter, the bodies remain at the vertices of an equilateral triangle, with any changes of size and orientation preserving that shape. The masses need not be equal.

For three equal masses, Chenciner and Montgomery (2000) proved the existence of a collision-free periodic solution in which all three bodies follow the same planar figure-eight path, one third of a period apart. Their proof uses the calculus of variations to establish the existence of this motion, rather than relying only on numerical integration.