Sitnikov problem

The Sitnikov problem is a restricted version of the three-body problem named after Russian mathematician Kirill Alexandrovitch Sitnikov that attempts to describe the movement of three celestial bodies due to their mutual gravitational attraction.

A special case of the Sitnikov problem was first discovered by the American scientist William Duncan MacMillan in 1911, but the problem as it currently stands wasn't discovered until 1961 by Sitnikov.

, which move in circular or elliptical Kepler orbits around their center of mass.

The origin of the system is at the focus of the primary bodies.

In order to derive the equation of motion in the case of circular orbits for the primary bodies, use that the total energy

is: After differentiating with respect to time, the equation becomes: This, according to Figure 1, is also true: Thus, the equation of motion is as follows: which describes an integrable system since it has one degree of freedom.

If on the other hand the primary bodies move in elliptical orbits then the equations of motion are where

Now the system has one-and-a-half degrees of freedom and is known to be chaotic.