On the problem of 𝑛 bodies

Abstract

Introduction. For the problem of three bodies, Sundmant established together with other results that if the angular momentum of the three bodies is not zero about every axis through the center of gravity of the system, the greatest of the three mutual distances will always exceed a specifiable constant depending upon the initial configuration of the bodies, and hence that triple collision is impossible. The problem was then considered from a different point of view by Birkhofft in his Chicago Colloquium lectures of 1920. He considered the case for which the angular momentum of the three bodies about every axis through the center of gravity of the system is not zero and for which the constant K appearing in the energy integral: T = U K, is (1) equal to or less than zero, and (2) greater than zero. Here T denotes the kinetic energy and U denotes the potential energy of the system. He showed for the first case that at least two if not all three of the mutual distances increase indefinitely as the time increases and decreases. For the second case, he showed if the motion of the three bodies is such that for some instant all three bodies approach sufficiently near to one another, that two of the mutual distances become infinite with the time while the third mutual distance remains less than a definite constant depending only upon the energy constant and the total mass of the system. After stating and proving various other results, he concluded by stating without formal proof that the results described above may be extended to the case of n bodies attracting one another according to the Newtonian law of force as well as to the case of n bodies attracting one another according to a more general law of force. The present paper has as its object the investigation of the conditions under which these extensions apply. The equations of motion and other fundamental relationships. We shall denote the n bodies (assumed to be particles) by Pi (i = 1, 2, * * *, n), and suppose them to have positive finite masses mi and real coordinates (xi, yi, zi). The distance from Pi to Pi will be denoted by ri2. We shall suppose that the bodies attract one another in such a way that there exists a potential function