  It was claimed recently that a low order post-Newtonian (PN) Lagrangian
formulation, which corresponds to the Euler-Lagrange equations up to an
infinite PN order, can be identical to a PN Hamiltonian formulation at the
infinite order from a theoretical point of view. This result is difficult to
check because in most cases one does not know what both the Euler-Lagrange
equations and the equivalent Hamiltonian are at the infinite order. However, no
difficulty exists for a special 1PN Lagrangian formulation of relativistic
circular restricted three-body problem, where both the Euler-Lagrange equations
and the equivalent Hamiltonian not only are expanded to all PN orders but also
have converged functions. Consequently, the analytical evidence supports this
claim. As far as numerical evidences are concerned, the Hamiltonian equivalent
to the Euler-Lagrange equations for the lower order Lagrangian requires that
they both be only at higher enough finite orders.
