Symplectic integrators for post-Newtonian Lagrangian dynamics

Abstract

In general, a truncation for the exact implicit equations of motion is necessary when quantitative research is carried out for the post-Newtonian Lagrangian dynamics of high nonlinearity. However, this truncation for equations of motion and the using of featureless numerical methods may lead to the essential gap between numerical and theoretical results because chaos amplifies the initial truncation error to an unacceptable level after long-term numerical integration. To provide highly accurate numerical solutions for post-Newtonian Lagrangian systems of compact objects, we investigate symplectic integrators incorporating the nontruncated strategy in this paper. By virtue of the canonical form of Hamiltonian dynamics, we propose an effective approach to construct symplectic integrators for the post-Newtonian Lagrangian dynamics, where an iteration procedure to obtain velocities from the generalized momentums is needed. That is, the post-Newtonian Lagrangian is numerically solved not in the Lagrangian frame but in the Hamiltonian frame. Numerical experiments with the post-Newtonian Lagrangian circular restricted three-body problem, and the Lagrangian spinning compact binaries at the second post-Newtonian order soundly support the efficiency and robustness of the proposed symplectic integrators. Finally, the mixed symplectic integrators are preferred because of their improved performance when the separation of one solvable dominate part with another perturbation part is possible for the equivalent Hamiltonian.