The mathematical problem of finding the velocities and positions of any number (n) of objects moving under their mutual gravitational attraction for any time in the past or future; also known as the many-body problem. It applies, for example, to the members of the Solar System and the members of a star cluster. As with the three-body problem, no general mathematical solution holding for all time has ever been found, but certain general results are known, true for any number of bodies. The centre of mass of the system of particles travels with constant velocity; the total energy of the system is constant; and the total angular momentum of the system does not change. With modern electronic computers, the bodies' velocities and positions can be calculated to any desired accuracy for finite lengths of time into the past or future but, for reasons such as round-off error (the error accumulating from slight initial inaccuracies when a large number of calculations are performed) and the chaotic nature of the problem, the accuracy deteriorates as the time increases.
Subjects: Astronomy and Astrophysics.