  Coherent or exact equations of motion for a post-Newtonian Lagrangian
formalism are the Euler-Lagrange equations without any terms truncated. They
naturally conserve energy {and} angular momentum. Doubling the phase-space
variables of positions and momenta in the coherent equations, we establish
extended phase-space symplectic-like integrators with the midpoint
permutations. The velocities should be solved iteratively from the algebraic
equations of the momenta defined by the Lagrangian during the course of
numerical integrations. It is shown numerically that a fourth-order extended
phase-space symplectic-like method exhibits good long-term stable error
behavior in energy {and} angular momentum, as a fourth-order implicit
symplectic method with a symmetric composition of three second-order implicit
midpoint rules or a fourth-order Gauss-Runge-Kutta implicit symplectic scheme
does. For given time step and integration time, the former method is superior
to the latter integrators in computational efficiency. {The extended
phase-space method is used to study the effects of the parameters and initial
conditions on the orbital dynamics of the coherent Euler-Lagrange equations for
a post-Newtonian circular restricted three-body problem. It is also applied to
trace the effects of the initial spin angles, initial separation and initial
orbital eccentricity on the dynamics of the coherent post-Newtonian
Euler-Lagrange equations of spinning compact binaries.
