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Equations of motion for a general relativistic post-Newtonian Lagrangianapproach mainly refer to acceleration equations, i.e. differential equations ofvelocities. They are directly from the Euler-Lagrangian equations, and usuallyhave higher-order terms truncated when they remain at the same post-Newtonianorder of the Lagrangian. In this sense, they are incoherent equations of theLagrangian and approximately conserve constants of motion in this system. Inthis paper, we show that the Euler-Lagrangian equations can also yield theequations of motion for consistency of the Lagrangian in the general case. Thecoherent equations are the \emph{differential} equations of generalized momentarather than those of the velocities, and have no terms truncated. Thevelocities 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 relativisticfields of compact objects as examples, we numerically evaluate the accuraciesof the constants of motion in the two sets of equations of motion. It isconfirmed that these accuracies well satisfy the theoretical need if the chosenintegrator can provide a high enough precision. The differences in thedynamical behavior of order and chaos between the two sets of equations arealso compared. Unlike the incoherent post-Newtonian Lagrangian equations ofmotion, the coherent ones can theoretically, strictly conserve all integrals insome post-Newtonian Lagrangian problems, and therefore are worth recommending.

The equations of motion for consistency of a post-Newtonian Lagrangian formulation