  In this paper, we focus on the set of geodesics rays of the Newtonian N-body
problem. We find that the limits of geodesic rays are also geodesic rays, hence
they are not dense in the space of initial conditions. As a result, there are
many motions whose domain is the half real line has non-negative total energy,
and they are not geodesic rays, the set of such motions has positive measure,
we think this set gives a large space to accommodate many solutions with "bad"
behaviors. As an application of weak KAM theory for N-body problem, we give a
brief proof of the existence of complete parabolic orbit starting from any
given initial position.
