  Using the canonical Arnowitt-Deser-Misner Hamiltonian formalism, a "first law
of mechanics" is established for binary systems of point masses moving along
generic stable bound (eccentric) orbits. This relationship is checked to hold
within the post-Newtonian approximation to general relativity, up to third
(3PN) order. Several applications are discussed, including the use of
gravitational self-force results to inform post-Newtonian theory and the
effective one-body model for eccentric-orbit compact binaries.
