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Using the canonical (ADM) Hamiltonian formalism, a ``first law of mechanics'' for binary point masses is presented, and is explicitly verified in the weak field perturbative limit. This ``first law'' provides a means of comparing predictions using different formalisms, valid for different parameter regimes, e.g. Post-Newtonian perturbation theory, numerical relativity or self-force calculations.
Alexandre Le Tiec (Wed,) studied this question.