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

TL;DR

This paper introduces extended phase-space symplectic-like integrators for post-Newtonian Euler-Lagrange equations, demonstrating their stability and efficiency in simulating complex orbital dynamics of spinning binaries.

Contribution

It develops a novel extended phase-space symplectic integrator framework for coherent post-Newtonian equations, improving long-term stability and computational efficiency.

Findings

Fourth-order method shows good long-term energy and angular momentum conservation.

The extended phase-space method effectively studies orbital dynamics and parameter effects.

The method outperforms traditional integrators in efficiency for given accuracy.

Abstract

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.