The existence of noncollision singularities in newtonian systems

In this paper we solve a long-standing problem in celestial mechanics proposed by Painleve and Poincare in the last century. The problem, which concerns the nature of the singularities in the n-body problem, asks whether there exists a noncollision singularity in the newtonian n-body problem? Here we give an affirmative answer to this problem by proving the existence of noncollision singularities in the 5-body problem. We consider n point-masses moving in a euclidean space W3. Let the mass of the ith particle be mi > 0, let its position be qj E R3 and let 4i E ]3 be its velocity. According to Newton's law,