Extended phase-space symplectic-like integrators for coherent post-Newtonian Euler-Lagrange equations

Coherent or exact equations of motion for a conservative post-Newtonian Lagrangian formalism are the Euler-Lagrange equations without any terms truncated. They naturally conserve energy and/or angular momentum. Doubling the phase-space variables of position 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 integration. It is shown numerically that a fourth-order extended phase-space symplectic-like method exhibits a good long-term stable error behavior in energy and/or angular momentum, as a fourth-order implicit symplectic method with a symmetric composition of three second-order implicit midpoint rules and a fourth-order Gauss-Runge-Kutta implicit symplectic scheme does. For a 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.