N -body 2PN Hamiltonian and numerical integration of the equations of motion

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 Arnowitt-Deser-Misner (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 the 2PN order using different methods and discuss several strategies for improving computational efficiency.