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Equivalence between the ADM-Hamiltonian and the harmonic-coordinates approaches to the third post-Newtonian dynamics of compact binaries

The third post-Newtonian approximation to the general relativistic dynamics of two point-mass systems has been recently derived by two independent groups, using different approaches, and different coordinate systems. By explicitly exhibiting the map between the variables used in the two approaches we prove their physical equivalence. Our map allows one to transfer all the known results of the Arnowitt-Deser-Misner (ADM) approach to the harmonic-coordinates one: in particular, it gives the value of the harmonic-coordinates Lagrangian, and the expression of the ten conserved quantities associated to global Poincaré invariance.