BURRAU'S PROBLEM OF THREE BODIES

Abstract

initially. The problem treated in this paper is to determine their subsequent motion. Because of the relation of the initial conditions to a Pythagoreani triangle, Burrau's problem may also be referred to as a Pythagorean problem of three bodies. That the motion of the particles takes place in the plane of the triangle is an immediate consequence of the initial conditions. Since the angular momentum of the system L at the beginning of the motion is zero, it will remain zero during the motion. The necessary but not sufficient condition for the simultaneous collision of all three particles is that L = 0 as shown for inistance by Wintner3 (pp. 253 and 265). This theorem, therefore, does not predict triple collisions in our case but allows the possibility of their occurrence. The other integrals of motion, that of the center of mass and of the energy, are applicable without any complication, but also without importance regarding the solution. Since triple collisions are not continuable singularities, the existence of the solution for the previously described initial conditions is in question. One practical approach to this problem of existence is nunierical experiments. Introducing variables which regularize binary collisions, nunmerical integrations are performed at several (increasing) levels of accuracy and the (hopefully convergent) results of these integrations are analyzed for the occurrence of triple collisions. If there appear only binary collisions, the solution is considered established. (3) This section is dedicated to the concept, the idea, and the definitionl of the solution. The existence of the solution is established once the occurrence of triple collisions is excluded. This is certainly the case when L # 0 because of the previously mentioned necessary condition for triple collisions. Unfortunately, in our problem L = 0, therefore the existence of the solution must be first established along the experimental lines indicated previously. Assuming then that the occurrence of triple collisions may be excluded, reference is made to Sundman's4 and Poincar6's5 work and to their representation of the solution in power series, convergent for all values of the time. Such definition (or existence) of the solution is of ques