  First laws of black hole mechanics, or thermodynamics, come in a variety of
different forms. In this paper, from a purely post-Newtonian (PN) analysis, we
obtain a first law for binary systems of point masses moving along an exactly
circular orbit. Our calculation is valid through 3PN order and includes, in
addition, the contributions of logarithmic terms at 4PN and 5PN orders. This
first law of binary point-particle mechanics is then derived from first
principles in general relativity, and analogies are drawn with the single and
binary black hole cases. Some consequences of the first law are explored for PN
spacetimes. As one such consequence, a simple relation between the PN binding
energy of the binary system and Detweiler's redshift observable is established.
Through it, we are able to determine with high precision the numerical values
of some previously unknown high order PN coefficients in the circular-orbit
binding energy. Finally, we propose new gauge invariant notions for the energy
and angular momentum of a particle in a binary system.
