Improbability of collisions in Newtonian gravitational systems

It is shown that the set of initial conditions leading to a collision in finite time has measure zero. 1. It is well known that binary collisions of point masses in a Newtonian gravitational system are improbable in the sense that the set of initial conditions leading to this catastrophe at some finite time has (Lebesgue volume) measure zero. One would expect the same to be true for multiple collisions if only for some sort of aesthetic reasoning—there seems to be a binary collision contained within a multiple collision. What is shown here is that this is indeed the case; that is, the set of initial conditions leading to collision in finite time has measure zero. This problem has gained attention in recent years with its inclusion in J. E. Littlewood's list of problems [2, Problem 13]. It must be emphasized that the fact that the force law is the inverse square force law plays a crucial role in the proof of this result. To see that this is not true for all force laws, let 21= 2 mKrf, where m¡ is the mass and r{ the position vector of the ith mass relative to the center of mass of the systems. It is well known (see [3], [6], [8], for example) that in the inverse p force law I=i3-p)T+ip-l)h. Here T is the kinetic energy, A is the total energy of the system and the dots denote differentiation with respect to time. Note that if p^3, l-¿ip-\)h. Integration yields IHp-i)ht2/2 + IiO)t + IiO). It follows that all initial conditions possessing negative A have the property that in finite time /—>0. Hence if the solution lasts long enough then the system will suffer a complete collapse. In particular, if n = 2 and A<0, there will be a collision. But the set of initial conditions yielding negative A has measure greater than zero. (Of course in the inverse square law the above is not true as for « = 2, negative energy leads in general to elliptic motion.) The reason that binary collisions are improbable in the inverse square law is that the system retains its analytic dependence on initial conditions at collision. (This comes from the fact that a binary collision can be regularized. See, for Received by the editors September 4, 1970 and, in revised form, January 19, 1971. AMS 1969 subject classifications. Primary 7034; Secondary 3440, 8500.