Equations of motion of self-gravitating $N$-body systems in the first post-Minkowskian approximation

We revisit the problem of the equations of motion of a system of $N$ self-interacting massive particles (without spins) in the first post-Minkowskian (1PM) approximation of general relativity. We write the equations of motion, gravitational field and associated conserved integrals of the motion in a form suitable for comparison with recently published post-Newtonian (PN) results at the 4PN order. We show that the Lagrangian associated with the equations of motion in harmonic coordinates is a generalized one, and compute all the terms linear in $G$ up to 5PN order. We discuss the Hamiltonian in the frame of the center of mass and exhibit a canonical transformation connecting it to previous results directly obtained with the Hamiltonian formalism of general relativity. Finally we recover the known result for the gravitational scattering angle of two particles at the 1PM order.