  We consider the scattering of $n$ classical particles interacting via pair
potentials, under the assumption that each pair potential is "long-range", i.e.
being of order ${\cal O}(r^{-\alpha})$ for some $\alpha >0$. We define and
focus on the "free region", the set of states leading to well-defined and
well-separated final states at infinity. As a first step, we prove the
existence of an explicit, global surface of section for the free region. This
surface of section is key to proving the smoothness of the map sending a point
to its final state and to establishing a forward conjugacy between the $n$-body
dynamics and free dynamics.
