  In this paper we describe optimal reduction for the system of two bodies in
$\mathbb{R}^3$ whose Hamiltonian is invariant under rotations and translations.
In doing this, we introduce parametrizations and charts which help giving
explicit expressions in order to deal with geometric and dynamical aspects of
the reduction process. For this system, the standard assumptions of the
Marsden-Weinstein reduction process are only partially satisfied while optimal
reduction can be readily applied, and we study a comparison between those two
reduction processes. We describe potential applications to the study of
Post-Newtonian Hamiltonian systems for binary systems in astronomy.
