On the Singularities and the Qualitative Behavior of the Solutions of the Problem of Central Forces

Introduction. The purpose of the present note is to investigate the motion of two bodies under the influence of their mutual attractions, when these latter satisfy certain specific conditions. By choosing the origin of the coordinate system so that it coincides with one of the particles, the motion may be reduced essentially to that of a single particle under a static central force. The force function, U, will in addition be subjected to the requirement that it be homogeneous of degree X (# 0) in the Cartesian coordinates x and y. In the particular case where A = 0 the latter requirement is replaced by the condition that the first partial derivatives US, Uy, be homogeneous in x, y of degree X 1 = -1. This leads in the former case to a force function U proportional to rX and in the latter to one proportional to log r, where the constant of proportionality is chosen so as to correspond to an attractive force.1 While a number of the results concerning the nature of the paths in various particular cases have long been known, the present paper is concerned primarily with an aspect of the subject which appears not to have been treated in detail hitherto, viz., the analytic character of the collisions, a subject which appears elementary at first glance, but which turns out to be complicated enough to involve a rather deep analytical apparatus; in particular, the proof of the non-existence of an analytic continuation in certain cases will be made to depend on Bohr's theory of almost-periodic functions of a complex variable. At the same time, it has seemed desirable to complete and systematize the description of all the various cases arising from force functionls of the above type, and in this connection the most striking result is that in the range -2 < X < 0 the situation is the same as for the Newtonian exponent X = -1; in the sense that the non-vanishing of the constant angular momentum, c, remains necessary and sufficient for the non-existence of collisions