Within the class of all metric theories of gravity, compact celestial bodies which themselves contribute nonnegligibly to the gravitational metric field generally will not move on geodesics of the metric field as test bodies do. For this reason, a post-Newtonian metric field expansion is not necessarily adequate to describe the general dynamics of two or more such celestial bodies interacting with each other gravitationally. Studies have, therefore, been initiated with the objective to develop, as a substitute, a post-Newtonian Lagrangian formalism for describing the relativistic dynamics of compact celestial bodies. The present paper treats the dynamics at the first post-Newtonian level (1/c-squared). However, the internal structure of the celestial bodies influences the dynamics at orders in 1/c-squared via the Lagrangian coupling parameters, which are nonperturbative.