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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.06961 |
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| _version_ | 1866917254710427648 |
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| author | Heinze, Felix M. Schäfer, Gerhard Brügmann, Bernd |
| author_facet | Heinze, Felix M. Schäfer, Gerhard Brügmann, Bernd |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The N-Body 2PN Hamiltonian and Numerical Integration of the Equations of Motion Heinze, Felix M. Schäfer, Gerhard Brügmann, Bernd General Relativity and Quantum Cosmology 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. |
| title | The N-Body 2PN Hamiltonian and Numerical Integration of the Equations of Motion |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2602.06961 |