Coherent post-Newtonian Lagrangian equations of motion

Equations of motion for a general relativistic post-Newtonian Lagrangian formalism 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 system. In this sense, they are incoherent Lagrangian equations of motion and approximately conserve constants of motion in this system. In fact, the Euler-Lagrangian equations can also yield the equations of motion for consistency of the Lagrangian in the general case. They are the differential equations of generalized momenta, which have no terms truncated during the derivation from this Lagrangian. The velocities are not integration variables, but they can be solved from the algebraic equations of the generalized momenta by means of an iterative method. In this way, the coherent post-Newtonian Lagrangian equations of motion are obtained. 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 are much better in the coherent equations than those in the incoherent ones. The differences in the dynamics 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 of motion in post-Newtonian Lagrangian dynamical problems and therefore are worth recommending.