Non-truncated strategy to exactly integrate the post-Newtonian Lagrangian circular restricted three-body problem

Abstract

In this study, we present a novel non-truncated strategy by accompanying the fixed-point iteration with traditional numerical integrators. The proposed non-truncated strategy aims to exactly integrate implicit motion equations that are directly derived from the Lagrangian of the post-Newtonian circular restricted three-body problem. In comparison with the commonly used truncated approach, which cannot exactly but approximately preserve the generalized Jacobian constant (or energy) of the original Lagrangian system, the proposed non-truncated strategy has been determined to preserve this constant well. In fact, the non-truncated strategy and the truncated approach have a difference at second post-Newtonian order. Based on Kolmogorov–Arnold–Moser theory, this difference from the truncation in the equations of motion may lead to destroying the orbital configuration, dynamical behavior of order and chaos, and conservation of the post-Newtonian circular restricted three-body problem. The non-truncated strategy proposed in this study can avoid all these drawbacks and provide highly reliable and accurate numerical solutions for the post-Newtonian Lagrangian dynamics. Finally, numerical results show that the non-truncated strategy can preserve the generalized Jacobian constant in the accuracy of {mathscr {O}}(10^{-12}), whereas the truncated approach at the first post-Newtonian (1PN) order only has an accuracy of {mathscr {O}}(10^{-3}). Moreover, several orbits are observed to be escaping from the bounded region in the 1PN truncated system via the truncated strategy, but these escaping orbits are unobserved via the non-truncated strategy.