Geodesic rays of the $N$-body problem

For the Newtonian N-body problem, we study the Jacobi-Maupertuis metric of the nonnegative energy levels. We show that the geodesic rays are expansive, that is to say, all the distances between the bodies must be divergent functions. More precisely, we prove that the evolution of such motions asymptotically decomposes into free particles and subsystems in completely parabolic expansion. The theorem applies in particular to the maximal characteristic curves of any given global viscosity solution of the stationary Hamilton-Jacobi equation $H(x,d_xu) = h$.