Introduction

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions.

The general n-body problem can be stated in the following way.

For each body i, with mass mi, let ci(t) be its trajectory. The path followed by an object moving through spacetrajectory in three dimensional space, where the parameter t is interpreted as time. Then the acceleration c‘‘(t) of each mass mi satisfies by the law of gravity:

The solutions of this system of differential equations give the positions as a function of time.

The Three Body Problem is the problem of investigating the behaviour of three mutually attracting bodies (such as the Sun, Earth and Moon) and the stability of their motion.

This problem is surprisingly difficult to solve, even in the so called restricted three-body problem, corresponding to the simple case of the three masses moving in a common plane.

Central to an understanding of spacecraft navigation is Newton 's inverse square law of gravity.

Using this law we can write down equations which describe the motion of the sun, the planets and any spacecraft flying about between them. If we simplify the solar system and consider only the sun and the Earth we can solve the equations analytically . That means we can find a simple solution which predicts exactly where they'll be at any point in time given information about their positions and velocities at some starting point. This problem is known as the two body problem and the solutions describe the familiar eliptical orbits of the planets known since the time of Kepler.

Unfortunately, when we add a third body to our equations of motion, such as a spacecraft lost in space between the Earth and the sun, we can no longer find an analytical solution. The equations are unsolvable. This problem is known as the three body problem of celestial mechanics and remains the subject of research today.

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).

Although Lagrange probably never dreamed of man made spacecraft being "parked" at the points that bear his name this is precisely how they are used today. SOHO was orbiting the sun near L 1 , a good place for a spacecraft that studies the sun because it is never in the Earth's shadow and yet it can always "see" Earth closeby to beam back its findings. Unfortunately, L 1 is only partially stable - objects placed there have a tendency to wander off if on-board rockets are not used to correct their position from time to time.