  In the present paper, using the first-order approximation of the $n$-body
Lagrangian (derived on the basis of the post-Newtonian gravitational theory of
Einstein, Infeld, and Hoffman), we explicitly write down the equations of
motion for the planar circular restricted three-body problem. Additionally,
with some simplified assumptions, we obtain two formulas for estimating the
values of the mass/distance and velocity/speed of light ratios appropriate for
a given post-Newtonian approximation. We show that the formulas derived in the
present study, lead to a good numerical conservation of the Jacobi constant and
allow for an approximate equivalence between the Lagrangian and Hamiltonian
approaches at the same post-Newtonian order. Accordingly, the dynamics of the
system is analyzed in terms of the Poincar\'e sections method and Lyapunov
exponents, finding that for specific values of the Jacobi constant the dynamics
can be either chaotic or regular. Our results suggest that the chaoticity of
the post-Newtonian system is slightly in- creased in comparison with its
Newtonian counterpart.
