  The dynamics of compact binary systems at the fourth post-Newtonian (4PN)
approximation of general relativity has been recently completed in a
self-consistent way. In this paper, we compute the ten Poincar\'e constants of
the motion and present the equations of motion in the frame of the center of
mass (CM), together with the corresponding CM Lagrangian, conserved energy and
conserved angular momentum. Next, we investigate the reduction of the CM
dynamics to the case of quasi-circular orbits. The non local (in time) tail
effect at the 4PN order is consistently included, as well as the relevant
radiation-reaction dissipative contributions to the energy and angular
momentum.
