HEXAGONbeta

Normal Cone Is All You Need: Unsoftened Symplectic Integration Of The Newtonian N-Body Problem Across Collisions In Finite Steps

A mix of human-written and AI-generated textHuman understanding: some partsmath.DS — Dynamical Systemsmath.AG — Algebraic Geometrymath.CA — Classical Analysis and ODEsmath.DG — Differential Geometrymath.MP — Mathematical Physicsmath.NA — Numerical Analysismath.SG — Symplectic Geometrycs.LO — Logic in Computer Sciencecs.MS — Mathematical Software

Contributed by Jeremy Nusa ↗

Version 1 / Nov 06, 2026 / CC BY-SA 4.0

Abstract

In the classical formulation of the Newtonian NN-body problem, symplectic geometry and collision singularities appear fundamentally irreconcilable: fixed-step symplectic integrators preserve the Poincaré integral invariants on smooth orbits only by softening the gravitational potential or breaking down at close encounters, whereas classical Sundman and Levi-Civita regularizations resolve collisions only by deforming the global time flow or altering the canonical phase-space structure. In this work, we show that this dichotomy is not an intrinsic obstruction of Hamiltonian mechanics, but an artifact of working on the unblown Cartesian configuration space. By lifting the dynamics to the standard cotangent bundle T∗Q[N]T^*\mathbf{Q}[N] of the real oriented Fulton–MacPherson compactification Q[N]=Bl⁡Π(N)(M/R3)\mathbf{Q}[N] = \operatorname{Bl}_{\Pi(N)}(M/\mathbb{R}^3) and applying the Calaque–Safronov shifted cotangent deformation to the normal cone DZQ[N]→A1\mathcal{D}_Z \mathbf{Q}[N] \to \mathbb{A}^1, we construct an exact, unsoftened (ε=0\varepsilon = 0) Hamiltonian splitting covered by the combinatorial atlas of (2N−3)!!(2N-3)!! rooted binary Jacobi trees. Along the normal-cone degeneration, the interaction decomposes canonically into commuting Kepler–Sundman leaf flows, synchronous anti-Zeno supernodes, and a scale-invariant quadrupolar-screened apex potential of global C1,1\mathcal{C}^{1,1} regularity. Endowing the exceptional collision divisors with McGehee's unitary elastic bounce involution (BC2=Id\mathcal{B}_C^2 = \mathrm{Id}) renders the leaf flows globally complete and time-reversible across binary collisions in a finite number of physical steps, thereby lifting the Sheng–Suzuki negative-timestep barrier for high-order symmetric compositions. In the bi-graded Poisson–Lie algebra Ln,m\mathcal{L}_{n,m} associated with this splitting, we establish a Universal Double Obstruction Theorem at order six: momentum-parity alternation prevents any pure configuration-space corrector from cancelling the leading secular error, while a non-vanishing Hall cohomology class in H1(DA,L5,2)H^1(D_A, \mathcal{L}_{5,2}) creates an incompressible O(ε2h4)\mathcal{O}(\varepsilon^2 h^4) quadratic tidal floor. We overcome the parity obstruction by introducing a torque-free cotangent corrector K4∈L4,1∩P1K_4 \in \mathcal{L}_{4,1} \cap \mathcal{P}_1, realized via a palindromic Lie conjugator Z4BM4(h)\mathcal{Z}_4^{\mathrm{BM4}}(h) that vanishes naturally as O(r7/2)\mathcal{O}(r^{7/2}) at collisions (scheme PBM4\mathrm{PBM}_4). Whenever the trajectory changes local Jacobi chart or dyadic step-size level, a non-commutative covariant gauge connection TA→B(PBM4)T_{A\to B}^{(\mathrm{PBM4})} closes the Čech 1-cocycle on the atlas groupoid and eliminates interfacial shadow-energy jumps. Accompanied by a complete, axiom-free Lean 4 formalization and validated up to 116-bit quadruple precision against the REBOUND integrator suite (IAS15, TRACE, MERCURIUS, and WHFast), the resulting architecture preserves all Galilean Lie–Bargmann invariants to machine precision across secular, collisional, and chaotic multi-body regimes.

Provenance statement

This monograph and its companion software archive were produced through a human-directed, AI-assisted research and engineering workflow using Google Antigravity (an agentic coding and mathematical research environment powered by frontier large language models), coupled with strict deterministic verification via interactive theorem proving and dual-precision scientific computing: 1. Conception, Architecture & Editorial Workflow: The core geometric thesis—replacing ad hoc gravitational softening and coordinate-chart switching with a canonical splitting on the shifted cotangent normal-cone deformation over the real Fulton–MacPherson compactification Q[N]—along with the mathematical architecture, problem formulation, and iterative validation trajectory, was directed and validated by the author. Google Antigravity was used throughout the project to assist with literature synthesis, symbolic Baker–Campbell–Hausdorff and Hall Lie algebra derivations, LaTeX manuscript drafting, code scaffolding, and multi-pass adversarial referee audits. 2. Mechanized Formal Verification (Lean 4 + Mathlib 4): To guarantee mathematical rigor and rule out algebraic or logical errors across the foundational definitions, lemmas, and theorems of Sections 1–5, 35 modular proof files (416 formal declarations, including 214 theorems and lemmas) and 5 self-contained chapter suites were formalized and machine-checked in Lean 4 (v4.35.0-rc2) using Mathlib 4. Automated '#print axioms' audits verify that the entire formalization compiles with 0 'sorry', 0 'sorryAx', and 0 custom axioms. 3. Dual-Precision Computational Validation & Reproducibility (Python / Numba / mpmath / REBOUND): All numerical integrators, micro-kernels, and celestial mechanics campaigns (Sections 3, 6, and 7) were implemented in two mirrored Python 3.11 engines—'normal_cone_nbody_float64' (IEEE-754 double precision with LLVM Numba '@njit(fastmath=False)' micro-kernels) and 'normal_cone_nbody_float128' (116-bit / 35-digit quadruple precision via 'mpmath'/'gmpy2')—and benchmarked head-to-head against 'REBOUND 5.2.2' (IAS15, WHFast, TRACE, MERCURIUS). The complete formal and numerical suite (330 unit tests, 14 certified campaign datasets, and automated GitHub Actions CI) is permanently archived at https://github.com/jeremynusa-hub/normal_cone_nbody3d (DOI: 10.5281/zenodo.23982968).

Tools used

Google
GeminiVersion Flash 3.8
Google
GeminiVersion Argon
Lean
LeanVersion 4
Google
Antigravity

References

  1. Sverre J. Aarseth, Gravitational NN -body simulations: Tools and algorithms , Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  2. Richard F. Arenstorf, Periodic solutions of the restricted three body problem representing analytic continuations of K eplerian elliptic motions , American Journal of Mathematics 85 (1963), no. 1, 27--35.
  3. Scott Axelrod and I. M. Singer, Chern-- S imons perturbation theory. II , Journal of Differential Geometry 39 (1994), no. 1, 173--213.
  4. Valentine Bargmann, On unitary ray representations of continuous groups , Annals of Mathematics 59 (1954), no. 1, 1--46.
  5. Richard H. Battin, An introduction to the mathematics and methods of astrodynamics , AIAA, Reston, VA, 1999.
  6. Giancarlo Benettin and Antonio Giorgilli, On the H amiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms , Journal of Statistical Physics 74 (1994), no. 5-6, 1117--1143.
  7. George D. Birkhoff, The restricted problem of three bodies , Rendiconti del Circolo Matematico di Palermo 39 (1915), 265--334.
  8. Sergio Blanes, Fernando Casas, Ariadna Farr é s, Jacques Laskar, Joseba Makazaga, and Ander Murua, New families of symplectic splitting methods for numerical integration in dynamical astronomy , Applied Numerical Mathematics 68 (2013), 58--72.
  9. Sergio Blanes, Fernando Casas, and Jes ú s Ros, Processing symplectic methods for near-integrable H amiltonian systems , Celestial Mechanics and Dynamical Astronomy 77 (2000), no. 1, 17--36.
  10. Sergio Blanes and P. C. Moan, Practical symplectic partitioned R unge-- K utta and R unge-- K utta-- N ystr ö m methods , Journal of Computational and Applied Mathematics 142 (2002), no. 2, 313--330.
Show all 85 referencesShow fewer references
  1. Tjarda Boekholt and Simon Portegies Zwart, On the reliability of NN -body simulations , Computational Astrophysics and Cosmology 2 (2015), 2.
  2. V. A. Bogomolov, Dynamics of the vorticity on a sphere , Fluid Dynamics 12 (1977), no. 6, 863--870.
  3. Dirk Brouwer, On the accumulation of errors in numerical integration , The Astronomical Journal 46 (1937), no. 1072, 149--153.
  4. Carl Burrau, Numerische B erechnung eines S pezialfalles des D reik ö rperproblems , Astronomische Nachrichten 195 (1913), no. 6, 113--118.
  5. Damien Calaque and Pavel Safronov, Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone , arXiv preprint arXiv:2407.08622 (accepted in American Journal of Mathematics) (2024).arXiv
  6. Ricardo Campos, Najib Idrissi, Pascal Lambrechts, and Thomas Willwacher, Configuration spaces of manifolds with boundary , Ast é risque 449 (2024), 1--134.
  7. P. J. Channell and C. Scovel, Integrators for L ie-- P oisson dynamical systems , Physica D: Nonlinear Phenomena 50 (1991), no. 1, 80--88.
  8. Alain Chenciner and Richard Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses , Annals of Mathematics 152 (2000), no. 3, 881--901.
  9. Corrado De Concini and Claudio Procesi, Wonderful models of subspace arrangements , Selecta Mathematica (New Series) 1 (1995), no. 3, 459--494.
  10. Ana Cannas da Silva, Lectures on symplectic geometry , Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, Heidelberg, 2001.
  11. Leonardo de Moura, Soonho Kong, Jeremy Avigad, Floris van Doorn, and Gabriel Ebner, The L ean theorem prover (system description) , Automated Deduction -- CADE-25, Lecture Notes in Computer Science, vol. 9195, Springer, 2015, pp. 378--388.
  12. Ariadna Farr é s, Jacques Laskar, Sergio Blanes, Fernando Casas, Joseba Makazaga, and Ander Murua, High precision symplectic integrators for the S olar S ystem , Celestial Mechanics and Dynamical Astronomy 116 (2013), no. 2, 141--174.
  13. William Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2, Springer-Verlag, Berlin, Heidelberg, 1984.
  14. William Fulton and Robert MacPherson, A compactification of configuration spaces , Annals of Mathematics 139 (1994), no. 1, 183--225.
  15. Zhong Ge and Jerrold E. Marsden, Lie-- P oisson H amilton-- J acobi theory and L ie-- P oisson integrators , Physics Letters A 133 (1988), no. 3, 134--139.
  16. Victor Guillemin, Eva Miranda, and Ana Rita Pires, Symplectic and P oisson geometry on bb -manifolds , Advances in Mathematics 264 (2014), 864--896.
  17. Ernst Hairer and Christian Lubich, The life-span of backward error analysis for numerical integrators , Numerische Mathematik 76 (1997), no. 4, 441--462.
  18. Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric numerical integration: Structure-preserving algorithms for ordinary differential equations , 2nd ed., Springer Series in Computational Mathematics, vol. 31, Springer-Verlag, Berlin, Heidelberg, 2006.
  19. Douglas C. Heggie, Binary evolution in stellar dynamics , Monthly Notices of the Royal Astronomical Society 173 (1975), no. 3, 729--787.
  20. David M. Hernandez and Walter Dehnen, A study of symplectic integrators for planetary system problems: error analysis and comparisons , Monthly Notices of the Royal Astronomical Society 468 (2017), no. 3, 2614--2636.
  21. — , Switching integrators reversibly in the astrophysical NN -body problem , Monthly Notices of the Royal Astronomical Society 522 (2023), no. 3, 4639--4648.
  22. George W. Hill, Researches in the lunar theory , American Journal of Mathematics 1 (1878), no. 1, 5--26.
  23. Carl Gustav Jacob Jacobi, Sur le mouvement d'un point et sur un cas particulier du probl è me des trois corps , Comptes Rendus de l'Acad é mie des Sciences 3 (1836), 59--61.
  24. — , Sur l' é limination des nœuds dans le probl è me des trois corps , Journal f ü r die reine und angewandte Mathematik (Crelles Journal) 26 (1843), 115--131.
  25. Gustav Kirchhoff, Vorlesungen ü ber mathematische physik: Mechanik , B. G. Teubner, Leipzig, 1876.
  26. Paul Kustaanheimo and Eduard Stiefel, Perturbation theory of K epler motion based on spinor regularization , Journal f ü r die reine und angewandte Mathematik 218 (1965), 204--219.
  27. Jacques Laskar, Frequency analysis for multi-dimensional systems. G lobal dynamics and diffusion , Physica D: Nonlinear Phenomena 67 (1993), no. 1--3, 257--281.
  28. Jacques Laskar and Philippe Robutel, High order symplectic integrators for perturbed H amiltonian systems , Celestial Mechanics and Dynamical Astronomy 80 (2001), no. 1, 39--62.
  29. Tullio Levi-Civita, Traiettorie singolari ed urti nel problema ristretto dei tre corpi , Annali di Matematica Pura ed Applicata (Serie III) 9 (1904), 1--32.
  30. — , Sur la r é gularisation du probl è me des trois corps , Acta Mathematica 42 (1920), 99--144.
  31. Tiger Lu, David M. Hernandez, and Hanno Rein, TRACE : a code for time-reversible astrophysical close encounters , Monthly Notices of the Royal Astronomical Society 533 (2024), no. 3, 3708--3723.
  32. Christian Marchal, The three-body problem , Studies in Astronautics, vol. 4, Elsevier, Amsterdam, 1990.
  33. Christian Marchal and Donald G. Saari, On the final evolution of the nn -body problem , Journal of Differential Equations 20 (1976), no. 1, 150--186.
  34. G. I. Marchuk, Some application of splitting-up methods to the solution of mathematical physics problems , Aplikace matematiky 13 (1968), no. 2, 103--132.
  35. Jean-Pierre Marco and David Sauzin, Stability and instability for G evrey quasi-convex near-integrable H amiltonian systems , Publications Math é matiques de l'IH É S 96 (2003), 199--275.
  36. Jerrold E. Marsden and Matthew West, Discrete mechanics and variational integrators , Acta Numerica 10 (2001), 357--514.
  37. Richard McGehee, Triple collision in the collinear three-body problem , Inventiones Mathematicae 27 (1974), no. 3, 191--227.
  38. Robert I. McLachlan, Explicit L ie-- P oisson integration and the E uler equations , Physical Review Letters 71 (1993), no. 19, 3043--3046.
  39. — , On the numerical integration of ordinary differential equations by symmetric composition methods , SIAM Journal on Scientific Computing 16 (1995), no. 1, 151--168.
  40. Richard B. Melrose, The A tiyah-- P atodi-- S inger index theorem , Research Notes in Mathematics, vol. 4, A K Peters/CRC Press, Wellesley, MA, 1993.
  41. Seppo Mikkola and Sverre J. Aarseth, An implementation of NN -body chain regularization , Celestial Mechanics and Dynamical Astronomy 57 (1993), no. 3, 439--459.
  42. Keith W. Myerscough and Jason Frank, Explicit, parallel P oisson integration of point vortices on the sphere , Journal of Computational and Applied Mathematics 304 (2016), 100--119.
  43. Dang Pham, Hanno Rein, and David S. Spiegel, A new timestep criterion for NN -body simulations , The Open Journal of Astrophysics 7 (2024), 1.
  44. Henri Poincar é , Les m é thodes nouvelles de la m é canique c é leste , Gauthier-Villars, Paris, 1892.
  45. Kevin P. Rauch and Matthew Holman, Dynamical chaos in the W isdom-- H olman integrator: origins and solutions , The Astronomical Journal 117 (1999), no. 2, 1087--1102.
  46. Hanno Rein, David M. Hernandez, Daniel Tamayo, Garett Brown, Emily Eckels, Emma Holmes, Michelle Lau, R é jean Leblanc, and Ari Silburt, Hybrid symplectic integrators for planetary dynamics , Monthly Notices of the Royal Astronomical Society 485 (2019), no. 4, 5490--5497.
  47. Hanno Rein and Shang-Fei Liu, REBOUND : an open-source multi-purpose NN -body code for collisional dynamics , Astronomy & Astrophysics 537 (2012), A128.
  48. Hanno Rein and David S. Spiegel, IAS15 : a fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits , Monthly Notices of the Royal Astronomical Society 446 (2015), no. 2, 1424--1437.
  49. Hanno Rein and Daniel Tamayo, WHFast : a fast and unbiased implementation of a symplectic W isdom-- H olman integrator for long-term gravitational simulations , Monthly Notices of the Royal Astronomical Society 452 (2015), no. 1, 376--388.
  50. Hanno Rein, Daniel Tamayo, and Garett Brown, High-order symplectic integrators for planetary dynamics and their implementation in REBOUND , Monthly Notices of the Royal Astronomical Society 489 (2019), no. 4, 4632--4640.
  51. Donald G. Saari, Improbability of collisions in N ewtonian gravitational systems , Transactions of the American Mathematical Society 162 (1971), 267--271.
  52. — , Improbability of collisions in N ewtonian gravitational systems. II , Transactions of the American Mathematical Society 181 (1973), 351--368.
  53. — , A global analysis of the gravitational nn-body problem , Lecture Notes in Mathematics 597 (1977), 441--445.
  54. Qin Sheng, Solving linear partial differential equations by exponential splitting , IMA Journal of Numerical Analysis 9 (1989), no. 2, 199--212.
  55. Carl Ludwig Siegel, Der D reierstoß , Annals of Mathematics 42 (1941), no. 1, 127--168.
  56. Robert D. Skeel and C. William Gear, Does variable step size ruin a symplectic integrator? , Physica D: Nonlinear Phenomena 60 (1992), no. 1-4, 311--313.
  57. Stephen Smale, Topology and mechanics. I , Inventiones Mathematicae 10 (1970), no. 4, 305--331.
  58. Mark Sofroniou and Giulia Spaletta, Derivation of symmetric composition constants for symmetric integrators , Optimization Methods and Software 20 (2005), no. 4-5, 597--613.
  59. Gilbert Strang, On the construction and comparison of difference schemes , SIAM Journal on Numerical Analysis 5 (1968), no. 3, 506--517.
  60. Karl Stumpff, Himmelsmechanik: B and I : D as Z weik ö rperproblem und die M ethoden der B ahnbestimmung , Deutscher Verlag der Wissenschaften, Berlin, 1959.
  61. — , Himmelsmechanik: B and II : D as D reik ö rperproblem , Deutscher Verlag der Wissenschaften, Berlin, 1965.
  62. Karl F. Sundman, M é moire sur le probl è me des trois corps , Acta Mathematica 36 (1912), 105--179.
  63. Masuo Suzuki, Fractal decomposition of exponential operators with applications to many-body problems and M onte C arlo simulations , Physics Letters A 146 (1990), no. 6, 319--323.
  64. — , General theory of fractal path integrals with applications to many-body theories and statistical physics , Journal of Mathematical Physics 32 (1991), no. 2, 400--407.
  65. Victor Szebehely, Theory of orbits: The restricted problem of three bodies , Academic Press, New York, 1967.
  66. Victor Szebehely and C. Frederick Peters, Complete solution of a general problem of three bodies , The Astronomical Journal 72 (1967), 876--883.
  67. H. F. Trotter, On the product of semi-groups of operators , Proceedings of the American Mathematical Society 10 (1959), no. 4, 545--551.
  68. Jean-Louis Verdier, Sp é cialisation des classes de C hern , Ast é risque 82--83 (1981), 149--159, Caract é ristique d'Euler--Poincar é (S é minaire E.N.S. 1978--1979); see also A. Douady and J.-L. Verdier, S é minaire de g é om é trie analytique , Ast é risque 36--37 (1976).
  69. J ö rg Waldvogel, A new regularization of the planar problem of three bodies , Celestial Mechanics 6 (1972), no. 2, 221--231.
  70. Hassler Whitney, Tangents to an analytic variety , Annals of Mathematics 81 (1965), no. 3, 496--549.
  71. Jack Wisdom, Symplectic test particle encounters: a comparison of methods , Monthly Notices of the Royal Astronomical Society 464 (2017), no. 2, 2350--2355.
  72. Jack Wisdom and Matthew Holman, Symplectic maps for the NN -body problem , The Astronomical Journal 102 (1991), 1528--1538.
  73. Jack Wisdom, Matthew Holman, and Jihad Touma, Symplectic correctors , Integration Algorithms and Classical Mechanics (Jerrold E. Marsden, George W. Patrick, and William F. Shadwick, eds.), Fields Institute Communications, vol. 10, American Mathematical Society, Providence, RI, 1996, pp. 217--244.
  74. Zhihong Xia, The existence of noncollision singularities in N ewtonian systems , Annals of Mathematics 135 (1992), no. 3, 411--468.
  75. Haruo Yoshida, Construction of higher order symplectic integrators , Physics Letters A 150 (1990), no. 5-7, 262--268.

Version history

  1. v1 · 196 pagesSubmitted by Jeremy NusaInitial depositCurrentNov 05, 2026