The N-Body 2PN Hamiltonian and Numerical Integration of the Equations of Motion

TL;DR

This paper derives a general N-body 2PN Hamiltonian expression, introduces numerical methods for evaluating complex integrals, and demonstrates the feasibility of high-precision numerical integration of equations of motion at 2PN order.

Contribution

It provides the first analytic form of the N-body 2PN Hamiltonian with a single integral term and shows how to evaluate it numerically for practical simulations.

Findings

Numerical evaluation of integrals achieves machine precision.

Full numerical integration of N-body 2PN equations is feasible.

Strategies for improving computational efficiency are discussed.

Abstract

To date, the second-order post-Newtonian (2PN) Hamiltonian has been known in closed analytic form only for systems of up to three point masses. In this paper, we present an analytic expression for the general $N$-body 2PN Hamiltonian in the ADM gauge up to a single integral term that, to our knowledge, has no known closed-form analytic solution. We show that the integrals appearing in the 2PN Hamiltonian can be evaluated numerically to machine precision, allowing for cross-validation against analytical results and enabling the full numerical computation of the $N$-body 2PN Hamiltonian. Furthermore, we demonstrate the practical feasibility of the numerical integration of the equations of motion for $N$ bodies at 2PN order using different methods and discuss several strategies for improving computational efficiency.