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

Felix M. Heinze, Gerhard Schäfer, and Bernd Brügmann

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Phys. Rev. D 113, 104066 – Published 28 May, 2026

DOI: https://doi.org/10.1103/9vhx-6wkk

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 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.

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References (81)

  1. M. E. Pati and C. M. Will, Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. 1. Foundations, Phys. Rev. D 62, 124015 (2000).
  2. T. Futamase and Y. Itoh, The post-Newtonian approximation for relativistic compact binaries, Living Rev. Relativity 10, 2 (2007).
  3. L. Blanchet, Post-Newtonian theory for gravitational waves, Living Rev. Relativity 27, 4 (2024).
  4. G. Schäfer and P. Jaranowski, Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries, Living Rev. Relativity 27, 2 (2024).
  5. M. Levi, Effective field theories of post-Newtonian gravity: A comprehensive review, Rep. Prog. Phys. 83, 075901 (2020).
  6. J. Blümlein, A. Maier, P. Marquard, and G. Schäfer, Gravity in binary systems at the fifth and sixth post-Newtonian order, Proc. Sci. LL2022 (2022) 012 [arXiv:2208.04552].
  7. T. Damour and A. Nagar, The effective one body description of the two-body problem, Fundam. Theor. Phys. 162, 211 (2011).
  8. T. Damour, Gravitational scattering, post-Minkowskian approximation and effective one-body theory, Phys. Rev. D 94, 104015 (2016).
  9. L. Barack and A. Pound, Self-force and radiation reaction in general relativity, Rep. Prog. Phys. 82, 016904 (2019).
  10. C. R. T. Jones and M. Solon, Scattering amplitudes and N-body post-Minkowskian Hamiltonians in general relativity and beyond, J. High Energy Phys. 02 (2022) 105.
  11. F. Pretorius, Evolution of binary black-hole spacetimes, Phys. Rev. Lett. 95, 121101 (2005).
  12. M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96, 111101 (2006).
  13. J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96, 111102 (2006).
  14. Y. Zlochower, M. Campanelli, and C. O. Lousto, Modeling gravitational recoil using numerical relativity, Classical Quantum Gravity 28, 114015 (2011).
  15. L. Baiotti and L. Rezzolla, Binary neutron star mergers: A review of Einstein’s richest laboratory, Rep. Prog. Phys. 80, 096901 (2017).
  16. B. Brügmann, Fundamentals of numerical relativity for gravitational wave sources, Science 361, 366 (2018).
  17. M. Shibata and K. Hotokezaka, Merger and mass ejection of neutron-star binaries, Annu. Rev. Nucl. Part. Sci. 69, 41 (2019).
  18. S. Bernuzzi, Neutron star merger remnants, Gen. Relativ. Gravit. 52, 108 (2020).
  19. K. Kyutoku, M. Shibata, and K. Taniguchi, Coalescence of black hole–neutron star binaries, Living Rev. Relativity 24, 5 (2021).
  20. P. T. H. Pang et al., An updated nuclear-physics and multi-messenger astrophysics framework for binary neutron star mergers, Nat. Commun. 14, 8352 (2023).
  21. S. F. P. Zwart, T. C. N. Boekholt, E. Por, A. S. Hamers, and S. L. W. McMillan, Chaos in self-gravitating many-body systems—Lyapunov time dependence of N and the influence of general relativity, Astron. Astrophys. 659, A86 (2022).
  22. P. Di Cintio and A. A. Trani, Partial suppression of chaos in relativistic three-body problems, Astron. Astrophys. 693, A53 (2025).
  23. C. O. Lousto and Y. Zlochower, Foundations of multiple-black-hole evolutions, Phys. Rev. D 77, 024034 (2008).
  24. M. Campanelli, C. O. Lousto, and Y. Zlochower, Close encounters of three black holes, Phys. Rev. D 77, 101501 (2008).
  25. P. Galaviz, B. Brügmann, and Z. Cao, Numerical evolution of multiple black holes with accurate initial data, Phys. Rev. D 82, 024005 (2010).
  26. S. Bai, Z.-J. Cao, W.-B. Han, C.-Y. Lin, H.-J. Yo, and J.-P. Yu, From binary black hole simulation to triple black hole simulation, J. Phys. Conf. Ser. 330, 012016 (2011).
  27. G. Ficarra, A. Ciarfella, and C. O. Lousto, Close encounter of three black holes revisited, Phys. Rev. D 108, 064045 (2023).
  28. G. Ficarra and C. O. Lousto, Close encounter of three black holes III, arXiv:2406.11985.
  29. J. Bamber, S. L. Shapiro, M. Ruiz, and A. Tsokaros, Evolution of a black hole cluster in full general relativity, Phys. Rev. D 112, 024046 (2025).
  30. F. M. Heinze, B. Brügmann, T. Dietrich, and I. Markin, Exploring the dynamics of general relativistic binary-single and binary-binary encounters of black holes, Phys. Rev. D 112, 064068 (2025).
  31. C.-K. Qiao, Y. Zheng, and Z.-J. Cao, A python interface for numerical relativity program AMSS-NCKU, arXiv:2509.21652.
  32. S. Mikkola and D. Merritt, Implementing few-body algorithmic regularization with post-Newtonian terms, Mon. Not. R. Astron. Soc. 135, 2398 (2008).
  33. S. Harfst, A. Gualandris, D. Merritt, and S. Mikkola, A hybrid N-body code incorporating algorithmic regularization and post-Newtonian forces, Mon. Not. R. Astron. Soc. 389, 2 (2008).
  34. S. Aarseth, Mergers and ejections of black holes in globular clusters, Mon. Not. R. Astron. Soc. 422, 841 (2012).
  35. C. L. Rodriguez, P. Amaro-Seoane, S. Chatterjee, and F. A. Rasio, Post-Newtonian dynamics in dense star clusters: Highly-eccentric, highly-spinning, and repeated binary black hole mergers, Phys. Rev. Lett. 120, 151101 (2018).
  36. A. Rantala, T. Naab, F. P. Rizzuto, M. Mannerkoski, C. Partmann, and K. Lautenschütz, BIFROST: Simulating compact subsystems in star clusters using a hierarchical fourth-order forward symplectic integrator code, Mon. Not. R. Astron. Soc. 522, 5180 (2023).
  37. A. Einstein, L. Infeld, and B. Hoffmann, The gravitational equations and the problem of motion, Ann. Math. 39, 65 (1938).
  38. C. Königsdörffer, G. Faye, and G. Schäfer, The binary black hole dynamics at the third-and-a-half post-Newtonian order in the ADM formalism, Phys. Rev. D 68, 044004 (2003).
  39. C. M. Will, Incorporating post-Newtonian effects in N-body dynamics, Phys. Rev. D 89, 044043 (2014); 91, 029902(E) (2015).
  40. G. Schäfer, Three-body Hamiltonian in general relativity, Phys. Lett. A 123, 336 (1987).
  41. C. O. Lousto and H. Nakano, Three-body equations of motion in successive post-Newtonian approximations, Classical Quantum Gravity 25, 195019 (2008).
  42. P. Galaviz and B. Brügmann, Characterization of the gravitational wave emission of three black holes, Phys. Rev. D 83, 084013 (2011).
  43. P. Galaviz, Stability and chaos of hierarchical three black hole configurations, Phys. Rev. D 84, 104038 (2011).
  44. M. Bonetti, F. Haardt, A. Sesana, and E. Barausse, Post-Newtonian evolution of massive black hole triplets in galactic nuclei—I. Numerical implementation and tests, Mon. Not. R. Astron. Soc. 461, 4419 (2016).
  45. Y.-Z. Chu, The N-body problem in general relativity up to the second post-Newtonian order from perturbative field theory, Phys. Rev. D 79, 044031 (2009).
  46. M. C. Miller and D. P. Hamilton, Four-body effects in globular cluster black hole coalescence, Astrophys. J. 576, 894 (2002).
  47. J. M. Fregeau, P. Cheung, S. F. Portegies Zwart, and F. A. Rasio, Stellar collisions during binary-binary and binary-single star interactions, Mon. Not. R. Astron. Soc. 352, 1 (2004).
  48. M. Zevin, J. Samsing, C. Rodriguez, C.-J. Haster, and E. Ramirez-Ruiz, Eccentric black hole mergers in dense star clusters: The role of binary-binary encounters, Astrophys. J. 871, 91 (2019).
  49. M. J. Valtonen, S. Mikkola, P. Heinamaki, and H. Valtonen, Slingshot ejections from clusters of three and four black holes, Astrophys. J. Suppl. Ser. 95, 69 (1994).
  50. A. S. Hamers, H. B. Perets, F. Antonini, and S. F. Portegies Zwart, Secular dynamics of hierarchical quadruple systems: The case of a triple system orbited by a fourth body, Mon. Not. R. Astron. Soc. 449, 4221 (2015).
  51. A. S. Hamers and D. Lai, Secular chaotic dynamics in hierarchical quadruple systems, with applications to hot jupiters in stellar binaries and triples, Mon. Not. R. Astron. Soc. 470, 1657 (2017).
  52. M. Safarzadeh, A. S. Hamers, A. Loeb, and E. Berger, Formation and merging of mass gap black holes in gravitational wave merger events from wide hierarchical quadruple systems, Astrophys. J. Lett. 888, L3 (2020).
  53. G. Fragione and B. Kocsis, Black hole mergers from quadruples, Mon. Not. R. Astron. Soc. 486, 4781 (2019).
  54. P. Vynatheya and A. S. Hamers, How important is secular evolution for black hole and neutron star mergers in 2+2 and 3+1 quadruple-star systems?, Astrophys. J. 926, 195 (2022).
  55. P. G. Breen and D. C. Heggie, Dynamical evolution of black hole sub-systems in idealised star clusters, Mon. Not. R. Astron. Soc. 432, 2779 (2013).
  56. G. Fragione, B. Kocsis, F. A. Rasio, and J. Silk, Repeated mergers, mass-gap black holes, and formation of intermediate-mass black holes in dense massive star clusters, Astrophys. J. 927, 231 (2022).
  57. B. Gaete, D. R. G. Schleicher, A. Lupi, B. Reinoso, M. Fellhauer, and M. C. Vergara, Supermassive black hole formation via collisions in black hole clusters, Astron. Astrophys. 690, A378 (2024).
  58. C. M. Will, Post-Newtonian effects in N-body dynamics: Conserved quantities in hierarchical triple systems, Classical Quantum Gravity 31, 244001 (2014).
  59. H. Yu, S. Ma, M. Giesler, and Y. Chen, Spin and eccentricity evolution in triple systems: From the Lidov-Kozai interaction to the final merger of the inner binary, Phys. Rev. D 102, 123009 (2020).
  60. H. Lim and C. L. Rodriguez, Relativistic three-body effects in hierarchical triples, Phys. Rev. D 102, 064033 (2020).
  61. H. Suzuki, P. Gupta, H. Okawa, and K.-i. Maeda, Post-Newtonian Kozai–Lidov mechanism and its effect on cumulative shift of periastron time of binary pulsar, Mon. Not. R. Astron. Soc. 500, 1645 (2020).
  62. A. Kuntz, Precession resonances in hierarchical triple systems, Phys. Rev. D 105, 024017 (2022).
  63. K.-i. Maeda, P. Gupta, and H. Okawa, Chaotic von Zeipel-Lidov-Kozai oscillations of a binary system around a rotating supermassive black hole, Phys. Rev. D 108, 123041 (2023).
  64. S. M. Kopeikin, General relativistic equations of binary motion for extended bodies with conservative corrections and radiation damping, Sov. Astron. 29, 516 (1985).
  65. T. Ohta, H. Okamura, K. Hiida, and T. Kimura, Higher order gravitational potential for many-body system, Prog. Theor. Phys. 51, 1220 (1974).
  66. T. Damour and G. Schäfer, Lagrangians for n point masses at the second post-Newtonian approximation of general relativity, Gen. Relativ. Gravit. 17, 879 (1985).
  67. M. E. Pati and C. M. Will, Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. 2. Two-body equations of motion to second post-Newtonian order, and radiation reaction to 3.5 post-Newtonian order, Phys. Rev. D 65, 104008 (2002).
  68. T. Ohta, H. Okamura, T. Kimura, and K. Hiida, Coordinate condition and higher order gravitational potential in canonical formalism, Prog. Theor. Phys. 51, 1598 (1974).
  69. T. Ohta, H. Okamura, T. Kimura, and K. Hiida, Physically acceptable solution of Einstein’s equation for many-body system, Prog. Theor. Phys. 50, 492 (1973).
  70. T. Hahn, cuba—A library for multidimensional numerical integration, Comput. Phys. Commun. 168, 78 (2005).
  71. J. Berntsen, T. O. Espelid, and A. Genz, An adaptive algorithm for the approximate calculation of multiple integrals, ACM Trans. Math. Softw. 17, 437 (1991).
  72. J. Berntsen, T. O. Espelid, and A. Genz, dcuhre: An adaptive multidimensional integration routine for a vector of integrals, ACM Trans. Math. Softw. 17, 452 (1991).
  73. W. Squire and G. Trapp, Using complex variables to estimate derivatives of real functions, SIAM Rev. 40, 110 (1998).
  74. C. Lubich, B. Walther, and B. Brügmann, Symplectic integration of post-Newtonian equations of motion with spin, Phys. Rev. D 81, 104025 (2010).
  75. J. Seyrich and G. Lukes-Gerakopoulos, A symmetric integrator for non-integrable Hamiltonian relativistic systems, Phys. Rev. D 86, 124013 (2012).
  76. J. Seyrich, Gauss collocation methods for efficient structure preserving integration of post-Newtonian equations of motion, Phys. Rev. D 87, 084064 (2013).
  77. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, Berlin, Heidelberg, 2006).
  78. F. M. Heinze, https://github.com/fmheinze/pn-nbody (2026), version 2026-02-06.
  79. J. Samsing, M. MacLeod, and E. Ramirez-Ruiz, The formation of eccentric compact binary inspirals and the role of gravitational wave emission in binary-single stellar encounters, Astrophys. J. 784, 71 (2014).
  80. https://www.gauss-centre.eu.
  81. https://www.lrz.de.

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Phys. Rev. D 113, 104066– Published 28 May, 2026
  • Received 10 February 2026
  • Accepted 27 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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