  In this paper, we revisit and complete our preceding work on the Fokker
Lagrangian describing the dynamics of compact binary systems at the fourth
post-Newtonian (4PN) order in harmonic coordinates. We clarify the impact of
the non-local character of the Fokker Lagrangian or the associated Hamiltonian
on both the conserved energy and the relativistic periastron precession for
circular orbits. We show that the non-locality of the action, due to the
presence of the tail effect at the 4PN order, gives rise to an extra
contribution to the conserved integral of energy with respect to the
Hamiltonian computed on shell, which was not taken into account in our previous
work. We also provide a direct derivation of the periastron advance by taking
carefully into account this non-locality. We then argue that the infra-red (IR)
divergences in the calculation of the gravitational part of the action are
problematic, which motivates us to introduce a second ambiguity parameter, in
addition to the one already assumed previously. After fixing these two
ambiguity parameters by requiring that the conserved energy and the
relativistic periastron precession for circular orbits are in agreement with
numerical and analytical gravitational self-force calculations, valid in the
limiting case of small mass ratio, we find that our resulting Lagrangian is
physically equivalent to the one obtained in the ADM Hamiltonian approach.
