1.1. The N-Body Problem

The fundamental mathematical problem considered throughout this book is a non continuum problem called the N-body problem, which is described in complete generality as follows. In cgs units and for i = 1, 2,..., N, let P _i of mass m _i be at , have velocity = ( v _i,x, v _i,y, v _i,z), and have acceleration = ( a _ix, a _iy, a _i, _Z) at time t ? 0. Let the positive distance between P _i and P _j, i ? j, be r _ij = r _ji ? 0. Let the force on P _i due to P _j be , so that the force depends only on the distance between P _i and P _j. Also, assume that the force on P _j due to P _i satisfies . Then, given the initial positions and velocities of all the P _i,i = 1, 2, 3,..., N, the general N-body problem is to determine the motion of the system if each P _i interacts with all the other P _j's in the system.

The prototype N-body problem was formulated in about 1900. In it the P _i were the Sun and the then known eight planets, and the force on each P _i was gravitational...