  The general relativistic corrections in the equations of motion and
associated energy of a binary system of point-like masses are derived at the
third post-Newtonian (3PN) order. The derivation is based on a post-Newtonian
expansion of the metric in harmonic coordinates at the 3PN approximation. The
metric is parametrized by appropriate non-linear potentials, which are
evaluated in the case of two point-particles using a Lorentzian version of an
Hadamard regularization which has been defined in previous works.
Distributional forms and distributional derivatives constructed from this
regularization are employed systematically. The equations of motion of the
particles are geodesic-like with respect to the regularized metric. Crucial
contributions to the acceleration are associated with the non-distributivity of
the Hadamard regularization and the violation of the Leibniz rule by the
distributional derivative. The final equations of motion at the 3PN order are
invariant under global Lorentz transformations, and admit a conserved energy
(neglecting the radiation reaction force at the 2.5PN order). However, they are
not fully determined, as they depend on one arbitrary constant, which reflects
probably a physical incompleteness of the point-mass regularization. The
results of this paper should be useful when comparing theory to the
observations of gravitational waves from binary systems in future detectors
VIRGO and LIGO.
