  In this paper we consider the 1D Green-Naghdi system. This system describes
the evolution of water waves over a flat bottom in the shallow water regime in
terms of the surface height $h$ and the horizontal velocity $u$. We give a
Lagrangian formulation of the 1D Green-Naghdi system on a Sobolev type
diffeomorphism group. As an application of this formulation we prove local
well-posedness for $(h,u)$ in the Sobolev space $(1+H^s(\R)) \times
H^{s+1}(\R),\; s > 1/2$. This improves the local well-posedness range for the
1D Green-Naghdi system.
