  We apply a dressed perturbation theory to better organize and economize the
computation of high orders of the 2-body effective action of an inspiralling
Post-Newtonian gravitating binary. We use the effective field theory approach
with the non-relativistic field decomposition (NRG fields). For that purpose we
develop quite generally the dressing theory of a non-linear classical field
theory coupled to point-like sources. We introduce dressed charges and
propagators, but unlike the quantum theory there are no dressed bulk vertices.
The dressed quantities are found to obey recursive integral equations which
succinctly encode parts of the diagrammatic expansion, and are the classical
version of the Schwinger-Dyson equations. Actually, the classical equations are
somewhat stronger since they involve only finitely many quantities, unlike the
quantum theory. Classical diagrams are shown to factorize exactly when they
contain non-linear world-line vertices, and we classify all the possible
topologies of irreducible diagrams for low loop numbers. We apply the dressing
program to our Post-Newtonian case of interest. The dressed charges consist of
the dressed energy-momentum tensor after a non-relativistic decomposition, and
we compute all dressed charges (in the harmonic gauge) appearing up to 2PN in
the 2-body effective action (and more). We determine the irreducible skeleton
diagrams up to 3PN and we employ the dressed charges to compute several terms
beyond 2PN.
