  Sticky particle solutions to the one-dimensional pressureless gas dynamics
equations can be constructed by a suitable metric projection onto the cone of
monotone maps, as was shown in recent work by Natile and Savar\'{e}. Their
proof uses a discrete particle approximation and stability properties for first
order differential inclusions. Here we give a more direct proof that relies on
a result by Haraux on the differentiability of metric projections. We apply the
same method also to the one-dimensional Euler-Poisson system, obtaining a new
proof for the global existence of weak solutions.
