Page "Orbit" Paragraph 46

Note that that while bound orbits around a point mass or around a spherical body with an Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects ( as caused, for example, by the slight oblateness of the Earth, or by relativistic effects, changing the gravitational field's behavior with distance ) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion.

The two-body solutions were published by Newton in Principia in 1687.

In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the three-body problem ; however, it converges too slowly to be of much use.

Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.