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

TL;DR

This work delivers an analytic expression for the general $N$-body 2PN Hamiltonian in the ADM gauge, isolating a single four-point integral $I^{ ext{ln}}_{ab;cd}$ that lacks a closed form. The authors implement and validate a numerical scheme (via the Cuba library) to evaluate all $2$PN integrals to machine precision, enabling accurate numerical integration of $N$-body trajectories at 2PN order. They develop efficient force and energy computation strategies and demonstrate practical orbit integrations for four-body systems, revealing that the four-point term $U^{ ext{TT}}_{(4)}$ is most relevant in close configurations but typically contributes only a few percent to the total energy and dynamics. The approach provides a robust framework for self-consistent 2PN simulations of general $N$-body systems, with potential for switching or approximating $U^{ ext{TT}}_{(4)}$ to trade accuracy for performance in large-$N$ contexts.

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.

Figures (5)

• Figure 1: Left: The true and estimated relative errors for the numerical evaluation of three different integrals for a random particle configuration using different numbers of integrand evaluations. Right: The absolute deviation of the value of $I^{\mathrm{ln}}_{ab;cd}$ from the final value with $10^8$ cubature integrand evaluations, indicating convergence to the final result.
• Figure 2: Trajectories of the four bodies in the binary-binary resonance interaction. The solid lines indicate the simulation including $U^{\mathrm{TT}}_{(4)}$, and the dashed lines the one without $U^{\mathrm{TT}}_{(4)}$. The dots indicate the final positions of the bodies (filled including $U^{\mathrm{TT}}_{(4)}$, hollow without $U^{\mathrm{TT}}_{(4)}$). Left: 3D view for the first $650M$ of the evolution with projections onto different planes to give a better sense of the locations in 3D space. Right: $xy$-projection for the full evolution until $t_{\mathrm{final}}=2500M$.
• Figure 3: The top left panel shows the contributions of $U^{\mathrm{TT}}_{(4)}$ relative to $H$ as well as the ratio between their gradients $\mathbf{\nabla}U^{\mathrm{TT}}_{(4)}$ and $\mathbf{\nabla}H$ with respect to the particle positions for the close binary-binary encounter. The bottom left panel highlights how the quantities describing the particle separation $D_{\mathrm{max}}$, $D_{\mathrm{avg}}$, and $R_{\mathrm{g}}$ defined in Equations (\ref{['eq:D_max']})-(\ref{['eq:R_g']}) vary accordingly. In the panel on the right, the relative errors in the conservation of energy, linear momentum, and angular momentum are plotted.
• Figure 4: Trajectories of the four bodies in the hierarchical binary-binary system. Left: $xy$-projection of the full trajectories. Right: $xy$-projection of the trajectories of the final outer orbit. The solid lines indicate the simulation including $U^{\mathrm{TT}}_{(4)}$, and the dashed lines the one without $U^{\mathrm{TT}}_{(4)}$. The gray solid and dashed lines indicate the center-of-mass motion of the two binaries. The dots indicate the final positions of the bodies (filled including $U^{\mathrm{TT}}_{(4)}$, hollow without $U^{\mathrm{TT}}_{(4)}$).
• Figure 5: The top left panel shows the contributions of $U^{\mathrm{TT}}_{(4)}$ relative to $H$ as well as the ratio between their gradients $\mathbf{\nabla}U^{\mathrm{TT}}_{(4)}$ and $\mathbf{\nabla}H$ with respect to the particle positions for the hierarchical binary-binary system. The bottom left panel highlights how the quantities describing the particle separation $D_{\mathrm{max}}$, $D_{\mathrm{avg}}$, and $R_{\mathrm{g}}$ defined in Equations (\ref{['eq:D_max']})-(\ref{['eq:R_g']}) vary accordingly. In the panel on the right, the relative errors in the conservation of energy, linear momentum, and angular momentum are plotted.