Singularities of the n-body problem. I

Little is known about the nature of the singularities of the n-body problem. While it is plausible to suppose that they are due to collisions, this has never been established, except when n = 2 or n = 3. In the general case the best that can be said at present is the fact, due to PAINLEV~ [5], that a singularity occurs at the time to if and only if the minimum of the mutual distances between pairs of particles approaches zero as the time t approaches to. In the present paper we shall investigate the problem of singularities due to collisions. We define a singularity at time to to be due to collisions i f as t ~ to each particle approaches a definite position in the inertial coordinate f rame. This means, in view of PAINLEV~'S theorem, that at least two particles approach the same point. In 1908 VON ZEIPEL [4] published a statement to the effect that if the system remains bounded as t--+ to, then a singularity at time t o is due to collisions. His proof is erroneous, and the assertion still stands as a conjecture. The purpose of the present paper is to obtain necessary and sufficient conditions for a singularity due to collisions. It will be supposed that the origin of coordinates is fixed at the center of mass, and that the singularity occurs as t ~ 0 +. The following notation will be used. The symbols m k, re, vk denote respectively the mass, position and velocity of the kth particle. We define further