Third post-Newtonian dynamics of compact binaries: Equations of motion in the center-of-mass frame

The equations of motion of compact binary systems and their associated Lagrangian formulation have been derived in previous works at the third post-Newtonian (3PN) approximation of general relativity in harmonic coordinates. In the present work we investigate the binary’s relative dynamics in the center-of-frame frame (center of mass located at the origin of the coordinates). We obtain the 3PN-accurate expressions of the center-of-mass positions and equations of the relative binary motion. We show that the equations derive from a Lagrangian (neglecting the radiation reaction), from which we deduce the conserved center-of-mass energy and angular momentum at the 3PN order. The harmonic-coordinates center-of-mass Lagrangian is equivalent, via a contact transformation of the particles’ variables, to the center-of-mass Hamiltonian in ADM coordinates that is known from the postNewtonian ADM-Hamiltonian formalism. As an application we investigate the dynamical stability of circular binary orbits at the 3PN order. On leave from Institut d’Astrophysique de Paris, groupe de Gravitation et Cosmologie (GRεCO, FRE 2435 du CNRS), 98 boulevard Arago, 75014 Paris, France.