  Equations of motion for a general relativistic post-Newtonian Lagrangian
approach mainly refer to acceleration equations, i.e. differential equations of
velocities. They are directly from the Euler-Lagrangian equations, and usually
have higher-order terms truncated when they remain at the same post-Newtonian
order of the Lagrangian. In this sense, they are incoherent equations of the
Lagrangian and approximately conserve constants of motion in this system. In
this paper, we show that the Euler-Lagrangian equations can also yield the
equations of motion for consistency of the Lagrangian in the general case. The
coherent equations are the \emph{differential} equations of generalized momenta
rather than those of the velocities, and have no terms truncated. The
velocities are not integration variables, but they can be solved from the
\emph{algebraic} equations of the generalized momenta with an iterative method.
Taking weak relativistic fields in the Solar System and strong relativistic
fields of compact objects as examples, we numerically evaluate the accuracies
of the constants of motion in the two sets of equations of motion. It is
confirmed that these accuracies well satisfy the theoretical need if the chosen
integrator can provide a high enough precision. The differences in the
dynamical behavior of order and chaos between the two sets of equations are
also compared. Unlike the incoherent post-Newtonian Lagrangian equations of
motion, the coherent ones can theoretically, strictly conserve all integrals in
some post-Newtonian Lagrangian problems, and therefore are worth recommending.
