  We derive the equations of motion for binary systems of compact bodies in the
post-Newtonian (PN) approximation to general relativity. Results are given
through 2PN order (order (v/c)^4 beyond Newtonian theory), and for
gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method
is based on a framework for direct integration of the relaxed Einstein
equations (DIRE) developed earlier, in which the equations of motion through
3.5PN order can be expressed in terms of Poisson-like potentials that are
generalizations of the instantaneous Newtonian gravitational potential, and in
terms of multipole moments of the system and their time derivatives. All
potentials are well defined and free of divergences associated with integrating
quantities over all space. Using a model of the bodies as spherical,
non-rotating fluid balls whose characteristic size s is small compared to the
bodies' separation r, we develop a method for carefully extracting only terms
that are independent of the parameter s, thereby ignoring tidal interactions,
spin effects, and internal self-gravity effects. Through 2.5PN order, the
resulting equations agree completely with those obtained by other methods; the
new 3.5PN back-reaction results are shown to be consistent with the loss of
energy and angular momentum via radiation to infinity.
