On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

TL;DR

This paper derives post-Newtonian equations for the circular restricted three-body problem, demonstrating that the Jacobi integral is approximately conserved and analyzing the system's dynamics, including chaos and regularity, using numerical methods.

Contribution

It provides explicit post-Newtonian equations of motion and formulas for parameter estimation, extending classical three-body problem analysis to relativistic corrections.

Findings

Good numerical conservation of the Jacobi constant

Approximate equivalence of Lagrangian and Hamiltonian approaches

Post-Newtonian chaos slightly exceeds Newtonian chaos

Abstract

In the present paper, using the first-order approximation of the $n$-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass/distance and velocity/speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to a good numerical conservation of the Jacobi constant and allow for an approximate equivalence between the Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Accordingly, the dynamics of the system is analyzed in terms of the Poincar\'e sections method and Lyapunov exponents, finding that for specific values of the Jacobi constant the dynamics can be either chaotic or regular. Our results suggest that the chaoticity of the post-Newtonian system is slightly in- creased in comparison with its Newtonian counterpart.