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We derive the conservative part of the Lagrangian and the energy of agravitationally bound two-body system at fourth post-Newtonian order, up toterms quadratic in the Newton constant. We also show that such terms arecompatible with Lorentz invariance and we write an ansatz for thecenter-of-mass position. The remaining terms carrying higher powers of theNewton constant are currently under investigation.

The dynamics of the gravitational two-body problem at fourth post-Newtonian order and at quadratic order in the Newton constant