The 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 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.

💡 Research Summary

This paper makes a significant breakthrough in the field of post-Newtonian (PN) celestial mechanics by presenting, for the first time, an analytic expression for the general N-body Hamiltonian at the second post-Newtonian (2PN) order. Prior to this work, a closed-form analytic 2PN Hamiltonian was only known for systems of up to three point masses. The key obstacle for extending to N≥4 bodies was the absence of an analytic solution for a specific three-dimensional integral term arising from the four-point correlation function within the 2PN static potential.

The authors derive the complete 2PN Hamiltonian in the ADM gauge for an arbitrary number of bodies. This Hamiltonian consists of the Newtonian (0PN), first PN (1PN), and second PN (2PN) parts. The 2PN part includes various potential terms proportional to G^3, the most complex of which is the transverse-traceless static potential U_TT. This potential decomposes into two-point, three-point, and four-point correlation functions based on how many distinct particle indices are involved. While the two- and three-point functions (U_TT(2), U_TT(3)) have known closed forms, the four-point function U_TT(4) for systems with four or more distinct bodies contained an intractable integral.

The core achievement of this paper is reducing U_TT(4) to an explicit analytic expression plus a single remaining integral term, denoted I_ln_ab;cd. The authors demonstrate that although this integral lacks a known closed-form algebraic solution, it is mathematically well-defined and, crucially, can be evaluated numerically to machine precision. They employ the deterministic, globally adaptive cubature algorithm Cuhre from the Cuba library to compute this and other integrals in the Hamiltonian. By successfully reproducing the known analytic results for the simpler correlation functions with high accuracy, they validate the robustness of their numerical integration approach.

With the Hamiltonian now computationally accessible, the paper outlines the practical numerical integration of the equations of motion for N-body systems at the 2PN order. The canonical equations are derived from the Hamiltonian H. Since the 2PN dynamics are conservative (radiation reaction enters at 2.5PN), the conservation of energy, linear momentum, and angular momentum provides a stringent test for the accuracy of the numerical integration scheme. The authors discuss strategies for improving computational efficiency, paving the way for practical simulations.

This work effectively bridges a gap between analytical relativity and numerical methods. By providing a means to compute the full 2PN Hamiltonian for N bodies, it enables, for the first time, complete and accurate simulations of the conservative dynamics of systems with four or more bodies at this relativistic order. This has profound implications for astrophysical modeling, allowing detailed studies of hierarchical systems, encounters involving multiple black holes or neutron stars, and the secular evolution of dense stellar systems where post-Newtonian corrections can significantly influence long-term behavior and stability.