  Building upon the Noether charge formalism of Iyer and Wald, we derive a
variational formula for spacetimes admitting a Killing vector field, for a
generic energy-momentum distribution with compact support. Applying this
general result to the particular case of a binary system of spinning compact
objects moving along an exactly circular orbit, modelled using the multipolar
gravitational skeleton formalism, we derive a first law of compact binary
mechanics at dipolar order. We prove the equivalence of this new result with
the canonical Hamiltonian first law previously derived for binary systems of
spinning compact objects, for spins colinear with the orbital angular momentum.
This paper paves the way to an extension of the first law of binary mechanics
to the next quadrupolar order, thereby accounting for the spin-induced and
tidally-induced deformability of the compact bodies.
