\ifx\pdfcompresslevel\undefined\else\pdfcompresslevel=1\fi
\ifx\pdfobjcompresslevel\undefined\else\pdfobjcompresslevel=0\fi
\ifx\XeTeXversion\undefined
  \pdfoutput=1
\fi
\documentclass[11pt,reqno,a4paper]{amsart}

% ==============================================================================
% Preamble Modular Inclusions
% ==============================================================================
\input{preamble/packages.tex}
\input{preamble/macros.tex}
\input{preamble/code_styles.tex}
\input{preamble/cross_referencing.tex}

% ==============================================================================
% Document Metadata
% ==============================================================================
\title[Normal Cone Is All You Need]{%
    NORMAL CONE IS ALL YOU NEED:\\
    UNSOFTENED SYMPLECTIC INTEGRATION OF THE NEWTONIAN\\
    $N$-BODY PROBLEM ACROSS COLLISIONS IN FINITE STEPS
}

\author{J\'er\'emy \textsc{Nusa}}
\date{}

\begin{document}

\begin{abstract}
In the classical formulation of the Newtonian $N$-body problem, symplectic geometry and collision singularities appear fundamentally irreconcilable: fixed-step symplectic integrators preserve the Poincar\'e 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 $T^*\mathbf{Q}[N]$ of the real oriented Fulton--MacPherson compactification $\mathbf{Q}[N] = \operatorname{Bl}_{\Pi(N)}(M/\mathbb{R}^3)$ and applying the Calaque--Safronov shifted cotangent deformation to the normal cone $\mathcal{D}_Z \mathbf{Q}[N] \to \mathbb{A}^1$, we construct an exact, unsoftened ($\varepsilon = 0$) Hamiltonian splitting covered by the combinatorial atlas of $(2N-3)!!$ 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 $\mathcal{C}^{1,1}$ regularity. Endowing the exceptional collision divisors with McGehee's unitary elastic bounce involution ($\mathcal{B}_C^2 = \mathrm{Id}$) 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 $\mathcal{L}_{n,m}$ associated with this splitting, we establish a \emph{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 $H^1(D_A, \mathcal{L}_{5,2})$ creates an incompressible $\mathcal{O}(\varepsilon^2 h^4)$ quadratic tidal floor. We overcome the parity obstruction by introducing a torque-free cotangent corrector $K_4 \in \mathcal{L}_{4,1} \cap \mathcal{P}_1$, realized via a palindromic Lie conjugator $\mathcal{Z}_4^{\mathrm{BM4}}(h)$ that vanishes naturally as $\mathcal{O}(r^{7/2})$ at collisions (scheme $\mathrm{PBM}_4$). Whenever the trajectory changes local Jacobi chart or dyadic step-size level, a non-commutative covariant gauge connection $T_{A\to B}^{(\mathrm{PBM4})}$ closes the \v{C}ech 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 \texttt{REBOUND} integrator suite (\texttt{IAS15}, \texttt{TRACE}, \texttt{MERCURIUS}, and \texttt{WHFast}), the resulting architecture preserves all Galilean Lie--Bargmann invariants to machine precision across secular, collisional, and chaotic multi-body regimes.

%
%
\end{abstract}\maketitle\vspace{1.2em}\begin{center}\zenodoshield\hspace{0.8em}\githubshield\end{center}\clearpage\thispagestyle{plain}\vspace*{0.28\textheight}\begin{center}\textbf{\large PREFACE}\\[1.2em]\textit{Author's Statement on Methodology and Editorial Voice}\end{center}\vspace{1em}\begin{quotation}I, the author, explicitly state that this theoretical research, geometric formalization, and manuscript drafting were carried out with the collaborative assistance of the artificial intelligence Gemini models (Flash 3.8 \& Argon). In this spirit, I employ the editorial pronoun ``we'' throughout this paper not as a conventional \emph{pluralis majestatis}, but as an inclusive denominator embracing the cumulative continuum of the author, the artificial intelligence collaborator, and the historical lineage of past authors and mathematicians whose foundational discoveries made this work possible.\end{quotation}\vfill\clearpage %
\iffalse
\end{abstract}

\maketitle

\clearpage

\fi \tableofcontents

\clearpage

% ==============================================================================
% Chapter 1: Introduction
% ==============================================================================
\input{chapters/ch01_introduction/sec01_background/content.tex}

\clearpage
% =========================================================================
% CHAPTER 2 : CALAQUE--SAFRONOV NORMAL CONE METHODOLOGY AND ALGORITHMIC RECIPES
% =========================================================================
\section{Calaque--Safronov Normal Cone Methodology and Algorithmic Recipes}
\label{ch:ch02_conormal_foundations}

\input{chapters/ch02_conormal_foundations/sec01_rees_deformation/content.tex}
\input{chapters/ch02_conormal_foundations/sec02_algorithmic_recipe/content.tex}
\input{chapters/ch02_conormal_foundations/sec03_retrospective_cases/content.tex}

\clearpage
% =========================================================================
% CHAPTER 3 : THE PLANAR THREE-BODY PROTOTYPE AND CLASSICAL REGULARIZATIONS
% =========================================================================
\section{The Planar Three-Body Prototype and Classical Regularizations}
\label{ch:ch03_three_body_paradigm}

\input{chapters/ch03_three_body_paradigm/sec01_pcr3bp_canonical/content.tex}
\input{chapters/ch03_three_body_paradigm/sec02_classical_regularizations/content.tex}
\input{chapters/ch03_three_body_paradigm/sec03_arenstorf_validation/content.tex}

\clearpage
% =========================================================================
% CHAPTER 4 : FULTON--MACPHERSON COMPACTIFICATION AND MULTISCALE SPLITTING
% =========================================================================
\section{Fulton--MacPherson Compactification and Multiscale Splitting}
\label{ch:ch04_fulton_macpherson_splitting}

\input{chapters/ch04_fulton_macpherson_splitting/sec01_whitney_lattice/content.tex}
\input{chapters/ch04_fulton_macpherson_splitting/sec02_screened_apex_potential/content.tex}
\input{chapters/ch04_fulton_macpherson_splitting/sec03_synchronous_supernodes/content.tex}

\clearpage
% =========================================================================
% CHAPTER 5 : SYMPLECTIC DISCRETIZATION AND HIGH-ORDER SCHEMES
% =========================================================================
\section{Symplectic Discretization and High-Order Schemes}
\label{ch:ch05_higher_order_symplectic}

\input{chapters/ch05_higher_order_symplectic/sec01_strang_propagator/content.tex}
\input{chapters/ch05_higher_order_symplectic/sec02_suzuki_barrier_mcgehee/content.tex}
\input{chapters/ch05_higher_order_symplectic/sec03_bm4_seven_stages/content.tex}
\input{chapters/ch05_higher_order_symplectic/sec04_bm6_eleven_stages/content.tex}
\input{chapters/ch05_higher_order_symplectic/sec05_symplectic_correctors/content.tex}
\input{chapters/ch05_higher_order_symplectic/sec06_dyadic_multirate_scheduling/content.tex}

\clearpage
% =========================================================================
% CHAPTER 6 : COMPUTATIONAL ARCHITECTURE AND ALGORITHMIC REALIZATION
% =========================================================================
\section{Computational Architecture and Algorithmic Realization of the Normal Cone Symplectic Integrator}
\label{ch:ch06_algorithmic_architecture}

\input{chapters/ch06_algorithmic_architecture/sec01_global_pipeline/content.tex}
\input{chapters/ch06_algorithmic_architecture/sec02_initialization_shadow/content.tex}
\input{chapters/ch06_algorithmic_architecture/sec03_dyadic_scheduler/content.tex}
\input{chapters/ch06_algorithmic_architecture/sec04_symplectic_subcycle/content.tex}
\input{chapters/ch06_algorithmic_architecture/sec05_atlas_transitions_diagnostics/content.tex}
\input{chapters/ch06_algorithmic_architecture/sec06_computational_optimizations/content.tex}


\clearpage
% =========================================================================
% CHAPTER 7 : NUMERICAL BENCHMARKS
% =========================================================================
\section{Numerical Benchmarks}
\label{ch:ch07_benchmarks_and_lean4}

\input{chapters/ch07_benchmarks_and_lean4/sec01_unsoftened_benchmarks/content.tex}
\input{chapters/ch07_benchmarks_and_lean4/sec02_long_term_stability/content.tex}
\input{chapters/ch07_benchmarks_and_lean4/sec03_rebound_head_to_head/content.tex}

\clearpage
% =========================================================================
% CHAPTER 8 : CONCLUSION AND OUTLOOK
% =========================================================================
\section{Conclusion and Outlook}
\label{ch:ch08_conclusion_outlook}
\label{sub:conclusion_outlook}

\input{chapters/ch08_conclusion/content.tex}

\clearpage
% ==============================================================================
% Appendices & Certified Code Artifacts
% ==============================================================================
\input{appendices/main_appendices.tex}

% ==============================================================================
% Bibliography
% ==============================================================================
\clearpage
\color{black}
\normalcolor
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{10}

\bibitem{aarseth2003gravitational}
Sverre~J. Aarseth, \emph{Gravitational {$N$}-body simulations: Tools and
  algorithms}, Cambridge Monographs on Mathematical Physics, Cambridge
  University Press, Cambridge, 2003.

\bibitem{arenstorf1963periodic}
Richard~F. Arenstorf, \emph{Periodic solutions of the restricted three body
  problem representing analytic continuations of {K}eplerian elliptic motions},
  American Journal of Mathematics \textbf{85} (1963), no.~1, 27--35.

\bibitem{axelrod1994chern}
Scott Axelrod and I.~M. Singer, \emph{Chern--{S}imons perturbation theory.
  {II}}, Journal of Differential Geometry \textbf{39} (1994), no.~1, 173--213.

\bibitem{bargmann1954unitary}
Valentine Bargmann, \emph{On unitary ray representations of continuous groups},
  Annals of Mathematics \textbf{59} (1954), no.~1, 1--46.

\bibitem{battin1999introduction}
Richard~H. Battin, \emph{An introduction to the mathematics and methods of
  astrodynamics}, AIAA, Reston, VA, 1999.

\bibitem{benettin1994hamiltonian}
Giancarlo Benettin and Antonio Giorgilli, \emph{On the {H}amiltonian
  interpolation of near-to-the identity symplectic mappings with application to
  symplectic integration algorithms}, Journal of Statistical Physics
  \textbf{74} (1994), no.~5-6, 1117--1143.

\bibitem{birkhoff1915}
George~D. Birkhoff, \emph{The restricted problem of three bodies}, Rendiconti
  del Circolo Matematico di Palermo \textbf{39} (1915), 265--334.

\bibitem{blanes2013optimized}
Sergio Blanes, Fernando Casas, Ariadna Farr{\'e}s, Jacques Laskar, Joseba
  Makazaga, and Ander Murua, \emph{New families of symplectic splitting methods
  for numerical integration in dynamical astronomy}, Applied Numerical
  Mathematics \textbf{68} (2013), 58--72.

\bibitem{blanes2000processing}
Sergio Blanes, Fernando Casas, and Jes{\'u}s Ros, \emph{Processing symplectic
  methods for near-integrable {H}amiltonian systems}, Celestial Mechanics and
  Dynamical Astronomy \textbf{77} (2000), no.~1, 17--36.

\bibitem{blanes2002practical}
Sergio Blanes and P.~C. Moan, \emph{Practical symplectic partitioned
  {R}unge--{K}utta and {R}unge--{K}utta--{N}ystr{\"o}m methods}, Journal of
  Computational and Applied Mathematics \textbf{142} (2002), no.~2, 313--330.

\bibitem{boekholt2015reliability}
Tjarda Boekholt and Simon~Portegies Zwart, \emph{On the reliability of
  {$N$}-body simulations}, Computational Astrophysics and Cosmology \textbf{2}
  (2015), 2.

\bibitem{bogomolov1977dynamics}
V.~A. Bogomolov, \emph{Dynamics of the vorticity on a sphere}, Fluid Dynamics
  \textbf{12} (1977), no.~6, 863--870.

\bibitem{brouwer1937measures}
Dirk Brouwer, \emph{On the accumulation of errors in numerical integration},
  The Astronomical Journal \textbf{46} (1937), no.~1072, 149--153.

\bibitem{burrau1913berechnung}
Carl Burrau, \emph{Numerische {B}erechnung eines {S}pezialfalles des
  {D}reik{\"o}rperproblems}, Astronomische Nachrichten \textbf{195} (1913),
  no.~6, 113--118.

\bibitem{calaque2024deformation}
Damien Calaque and Pavel Safronov, \emph{Shifted cotangent bundles, symplectic
  groupoids and deformation to the normal cone}, arXiv preprint
  arXiv:2407.08622 (accepted in American Journal of Mathematics) (2024).

\bibitem{campos2024configuration}
Ricardo Campos, Najib Idrissi, Pascal Lambrechts, and Thomas Willwacher,
  \emph{Configuration spaces of manifolds with boundary}, Ast{\'e}risque
  \textbf{449} (2024), 1--134.

\bibitem{channell1991integrators}
P.~J. Channell and C.~Scovel, \emph{Integrators for {L}ie--{P}oisson dynamical
  systems}, Physica D: Nonlinear Phenomena \textbf{50} (1991), no.~1, 80--88.

\bibitem{chenciner2000remarkable}
Alain Chenciner and Richard Montgomery, \emph{A remarkable periodic solution of
  the three-body problem in the case of equal masses}, Annals of Mathematics
  \textbf{152} (2000), no.~3, 881--901.

\bibitem{deconcini1995wonderful}
Corrado~De Concini and Claudio Procesi, \emph{Wonderful models of subspace
  arrangements}, Selecta Mathematica (New Series) \textbf{1} (1995), no.~3,
  459--494.

\bibitem{cannas2001lectures}
Ana~Cannas da~Silva, \emph{Lectures on symplectic geometry}, Lecture Notes in
  Mathematics, vol. 1764, Springer-Verlag, Berlin, Heidelberg, 2001.

\bibitem{moura2015lean}
Leonardo de~Moura, Soonho Kong, Jeremy Avigad, Floris van Doorn, and Gabriel
  Ebner, \emph{The {L}ean theorem prover (system description)}, Automated
  Deduction -- CADE-25, Lecture Notes in Computer Science, vol. 9195, Springer,
  2015, pp.~378--388.

\bibitem{farres2013new_families}
Ariadna Farr{\'e}s, Jacques Laskar, Sergio Blanes, Fernando Casas, Joseba
  Makazaga, and Ander Murua, \emph{High precision symplectic integrators for
  the {S}olar {S}ystem}, Celestial Mechanics and Dynamical Astronomy
  \textbf{116} (2013), no.~2, 141--174.

\bibitem{fulton1984intersection}
William Fulton, \emph{Intersection theory}, Ergebnisse der Mathematik und ihrer
  Grenzgebiete, vol.~2, Springer-Verlag, Berlin, Heidelberg, 1984.

\bibitem{fulton1994compactification}
William Fulton and Robert MacPherson, \emph{A compactification of configuration
  spaces}, Annals of Mathematics \textbf{139} (1994), no.~1, 183--225.

\bibitem{ge1988lie}
Zhong Ge and Jerrold~E. Marsden, \emph{Lie--{P}oisson {H}amilton--{J}acobi
  theory and {L}ie--{P}oisson integrators}, Physics Letters A \textbf{133}
  (1988), no.~3, 134--139.

\bibitem{guillemin2014symplectic}
Victor Guillemin, Eva Miranda, and Ana~Rita Pires, \emph{Symplectic and
  {P}oisson geometry on {$b$}-manifolds}, Advances in Mathematics \textbf{264}
  (2014), 864--896.

\bibitem{hairer1997life}
Ernst Hairer and Christian Lubich, \emph{The life-span of backward error
  analysis for numerical integrators}, Numerische Mathematik \textbf{76}
  (1997), no.~4, 441--462.

\bibitem{hairer2006geometric}
Ernst Hairer, Christian Lubich, and Gerhard Wanner, \emph{Geometric numerical
  integration: Structure-preserving algorithms for ordinary differential
  equations}, 2nd ed., Springer Series in Computational Mathematics, vol.~31,
  Springer-Verlag, Berlin, Heidelberg, 2006.

\bibitem{heggie1975binary}
Douglas~C. Heggie, \emph{Binary evolution in stellar dynamics}, Monthly Notices
  of the Royal Astronomical Society \textbf{173} (1975), no.~3, 729--787.

\bibitem{hernandez2017symplectic}
David~M. Hernandez and Walter Dehnen, \emph{A study of symplectic integrators
  for planetary system problems: error analysis and comparisons}, Monthly
  Notices of the Royal Astronomical Society \textbf{468} (2017), no.~3,
  2614--2636.

\bibitem{hernandez2023reversibility}
\bysame, \emph{Switching integrators reversibly in the astrophysical {$N$}-body
  problem}, Monthly Notices of the Royal Astronomical Society \textbf{522}
  (2023), no.~3, 4639--4648.

\bibitem{hill1878researches}
George~W. Hill, \emph{Researches in the lunar theory}, American Journal of
  Mathematics \textbf{1} (1878), no.~1, 5--26.

\bibitem{jacobi1836}
Carl Gustav~Jacob Jacobi, \emph{Sur le mouvement d'un point et sur un cas
  particulier du probl{\`e}me des trois corps}, Comptes Rendus de
  l'Acad{\'e}mie des Sciences \textbf{3} (1836), 59--61.

\bibitem{jacobi1843elimination}
\bysame, \emph{Sur l'{\'e}limination des n{\oe}uds dans le probl{\`e}me des
  trois corps}, Journal f{\"u}r die reine und angewandte Mathematik (Crelles
  Journal) \textbf{26} (1843), 115--131.

\bibitem{kirchhoff1876}
Gustav Kirchhoff, \emph{Vorlesungen {\"u}ber mathematische physik: Mechanik},
  B. G. Teubner, Leipzig, 1876.

\bibitem{kustaanheimo1965perturbation}
Paul Kustaanheimo and Eduard Stiefel, \emph{Perturbation theory of {K}epler
  motion based on spinor regularization}, Journal f{\"u}r die reine und
  angewandte Mathematik \textbf{218} (1965), 204--219.

\bibitem{laskar1993frequency}
Jacques Laskar, \emph{Frequency analysis for multi-dimensional systems.
  {G}lobal dynamics and diffusion}, Physica D: Nonlinear Phenomena \textbf{67}
  (1993), no.~1--3, 257--281.

\bibitem{laskar2001large}
Jacques Laskar and Philippe Robutel, \emph{High order symplectic integrators
  for perturbed {H}amiltonian systems}, Celestial Mechanics and Dynamical
  Astronomy \textbf{80} (2001), no.~1, 39--62.

\bibitem{levicivita1904}
Tullio Levi-Civita, \emph{Traiettorie singolari ed urti nel problema ristretto
  dei tre corpi}, Annali di Matematica Pura ed Applicata (Serie III) \textbf{9}
  (1904), 1--32.

\bibitem{levicivita1920regularisation}
\bysame, \emph{Sur la r{\'e}gularisation du probl{\`e}me des trois corps}, Acta
  Mathematica \textbf{42} (1920), 99--144.

\bibitem{lu2024trace}
Tiger Lu, David~M. Hernandez, and Hanno Rein, \emph{{TRACE}: a code for
  time-reversible astrophysical close encounters}, Monthly Notices of the Royal
  Astronomical Society \textbf{533} (2024), no.~3, 3708--3723.

\bibitem{marchal1990threebody}
Christian Marchal, \emph{The three-body problem}, Studies in Astronautics,
  vol.~4, Elsevier, Amsterdam, 1990.

\bibitem{marchal1976general}
Christian Marchal and Donald~G. Saari, \emph{On the final evolution of the
  {$n$}-body problem}, Journal of Differential Equations \textbf{20} (1976),
  no.~1, 150--186.

\bibitem{marchuk1968applications}
G.~I. Marchuk, \emph{Some application of splitting-up methods to the solution
  of mathematical physics problems}, Aplikace matematiky \textbf{13} (1968),
  no.~2, 103--132.

\bibitem{marco2003stability}
Jean-Pierre Marco and David Sauzin, \emph{Stability and instability for
  {G}evrey quasi-convex near-integrable {H}amiltonian systems}, Publications
  Math{\'e}matiques de l'IH{\'E}S \textbf{96} (2003), 199--275.

\bibitem{marsden2001discrete}
Jerrold~E. Marsden and Matthew West, \emph{Discrete mechanics and variational
  integrators}, Acta Numerica \textbf{10} (2001), 357--514.

\bibitem{mcgehee1974triple}
Richard McGehee, \emph{Triple collision in the collinear three-body problem},
  Inventiones Mathematicae \textbf{27} (1974), no.~3, 191--227.

\bibitem{mclachlan1993explicit}
Robert~I. McLachlan, \emph{Explicit {L}ie--{P}oisson integration and the
  {E}uler equations}, Physical Review Letters \textbf{71} (1993), no.~19,
  3043--3046.

\bibitem{mclachlan1995composition}
\bysame, \emph{On the numerical integration of ordinary differential equations
  by symmetric composition methods}, SIAM Journal on Scientific Computing
  \textbf{16} (1995), no.~1, 151--168.

\bibitem{melrose1993atiyah}
Richard~B. Melrose, \emph{The {A}tiyah--{P}atodi--{S}inger index theorem},
  Research Notes in Mathematics, vol.~4, A K Peters/CRC Press, Wellesley, MA,
  1993.

\bibitem{mikkola1993implementation}
Seppo Mikkola and Sverre~J. Aarseth, \emph{An implementation of {$N$}-body
  chain regularization}, Celestial Mechanics and Dynamical Astronomy
  \textbf{57} (1993), no.~3, 439--459.

\bibitem{myerscough2016explicit}
Keith~W. Myerscough and Jason Frank, \emph{Explicit, parallel {P}oisson
  integration of point vortices on the sphere}, Journal of Computational and
  Applied Mathematics \textbf{304} (2016), 100--119.

\bibitem{pham2024new}
Dang Pham, Hanno Rein, and David~S. Spiegel, \emph{A new timestep criterion for
  {$N$}-body simulations}, The Open Journal of Astrophysics \textbf{7} (2024),
  1.

\bibitem{poincare1892}
Henri Poincar{\'e}, \emph{Les m{\'e}thodes nouvelles de la m{\'e}canique
  c{\'e}leste}, Gauthier-Villars, Paris, 1892.

\bibitem{rauch1999dynamical}
Kevin~P. Rauch and Matthew Holman, \emph{Dynamical chaos in the
  {W}isdom--{H}olman integrator: origins and solutions}, The Astronomical
  Journal \textbf{117} (1999), no.~2, 1087--1102.

\bibitem{rein2019mercurius}
Hanno Rein, David~M. Hernandez, Daniel Tamayo, Garett Brown, Emily Eckels, Emma
  Holmes, Michelle Lau, R{\'e}jean Leblanc, and Ari Silburt, \emph{Hybrid
  symplectic integrators for planetary dynamics}, Monthly Notices of the Royal
  Astronomical Society \textbf{485} (2019), no.~4, 5490--5497.

\bibitem{rein2012rebound}
Hanno Rein and Shang-Fei Liu, \emph{{REBOUND}: an open-source multi-purpose
  {$N$}-body code for collisional dynamics}, Astronomy \& Astrophysics
  \textbf{537} (2012), A128.

\bibitem{rein2015ias15}
Hanno Rein and David~S. Spiegel, \emph{{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
  \textbf{446} (2015), no.~2, 1424--1437.

\bibitem{rein2015whfast}
Hanno Rein and Daniel Tamayo, \emph{{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
  \textbf{452} (2015), no.~1, 376--388.

\bibitem{rein2019highorder}
Hanno Rein, Daniel Tamayo, and Garett Brown, \emph{High-order symplectic
  integrators for planetary dynamics and their implementation in {REBOUND}},
  Monthly Notices of the Royal Astronomical Society \textbf{489} (2019), no.~4,
  4632--4640.

\bibitem{saari1971improbability}
Donald~G. Saari, \emph{Improbability of collisions in {N}ewtonian gravitational
  systems}, Transactions of the American Mathematical Society \textbf{162}
  (1971), 267--271.

\bibitem{saari1973improbability}
\bysame, \emph{Improbability of collisions in {N}ewtonian gravitational
  systems. {II}}, Transactions of the American Mathematical Society
  \textbf{181} (1973), 351--368.

\bibitem{saari1977singularities}
\bysame, \emph{A global analysis of the gravitational $n$-body problem},
  Lecture Notes in Mathematics \textbf{597} (1977), 441--445.

\bibitem{sheng1989solving}
Qin Sheng, \emph{Solving linear partial differential equations by exponential
  splitting}, IMA Journal of Numerical Analysis \textbf{9} (1989), no.~2,
  199--212.

\bibitem{siegel1941der}
Carl~Ludwig Siegel, \emph{Der {D}reiersto{\ss}}, Annals of Mathematics
  \textbf{42} (1941), no.~1, 127--168.

\bibitem{skeel1992does}
Robert~D. Skeel and C.~William Gear, \emph{Does variable step size ruin a
  symplectic integrator?}, Physica D: Nonlinear Phenomena \textbf{60} (1992),
  no.~1-4, 311--313.

\bibitem{smale1970topology}
Stephen Smale, \emph{Topology and mechanics. {I}}, Inventiones Mathematicae
  \textbf{10} (1970), no.~4, 305--331.

\bibitem{sofroniou2005derivation}
Mark Sofroniou and Giulia Spaletta, \emph{Derivation of symmetric composition
  constants for symmetric integrators}, Optimization Methods and Software
  \textbf{20} (2005), no.~4-5, 597--613.

\bibitem{strang1968construction}
Gilbert Strang, \emph{On the construction and comparison of difference
  schemes}, SIAM Journal on Numerical Analysis \textbf{5} (1968), no.~3,
  506--517.

\bibitem{stumpff1959himmelsmechanik}
Karl Stumpff, \emph{Himmelsmechanik: {B}and {I}: {D}as {Z}weik{\"o}rperproblem
  und die {M}ethoden der {B}ahnbestimmung}, Deutscher Verlag der
  Wissenschaften, Berlin, 1959.

\bibitem{stumpff1962himmelsmechanik}
\bysame, \emph{Himmelsmechanik: {B}and {II}: {D}as {D}reik{\"o}rperproblem},
  Deutscher Verlag der Wissenschaften, Berlin, 1965.

\bibitem{sundman1912memoire}
Karl~F. Sundman, \emph{M{\'e}moire sur le probl{\`e}me des trois corps}, Acta
  Mathematica \textbf{36} (1912), 105--179.

\bibitem{suzuki1990fractal}
Masuo Suzuki, \emph{Fractal decomposition of exponential operators with
  applications to many-body problems and {M}onte {C}arlo simulations}, Physics
  Letters A \textbf{146} (1990), no.~6, 319--323.

\bibitem{suzuki1991general}
\bysame, \emph{General theory of fractal path integrals with applications to
  many-body theories and statistical physics}, Journal of Mathematical Physics
  \textbf{32} (1991), no.~2, 400--407.

\bibitem{szebehely1967theory}
Victor Szebehely, \emph{Theory of orbits: The restricted problem of three
  bodies}, Academic Press, New York, 1967.

\bibitem{szebehely1967complete}
Victor Szebehely and C.~Frederick Peters, \emph{Complete solution of a general
  problem of three bodies}, The Astronomical Journal \textbf{72} (1967),
  876--883.

\bibitem{trotter1959product}
H.~F. Trotter, \emph{On the product of semi-groups of operators}, Proceedings
  of the American Mathematical Society \textbf{10} (1959), no.~4, 545--551.

\bibitem{verdier1976specialisation}
Jean-Louis Verdier, \emph{Sp{\'e}cialisation des classes de {C}hern},
  Ast{\'e}risque \textbf{82--83} (1981), 149--159, Caract{\'e}ristique
  d'Euler--Poincar{\'e} (S{\'e}minaire E.N.S. 1978--1979); see also A.~Douady
  and J.-L.~Verdier, \emph{S{\'e}minaire de g{\'e}om{\'e}trie analytique},
  Ast{\'e}risque 36--37 (1976).

\bibitem{waldvogel1972new}
J{\"o}rg Waldvogel, \emph{A new regularization of the planar problem of three
  bodies}, Celestial Mechanics \textbf{6} (1972), no.~2, 221--231.

\bibitem{whitney1965tangents}
Hassler Whitney, \emph{Tangents to an analytic variety}, Annals of Mathematics
  \textbf{81} (1965), no.~3, 496--549.

\bibitem{wisdom2017symplectic}
Jack Wisdom, \emph{Symplectic test particle encounters: a comparison of
  methods}, Monthly Notices of the Royal Astronomical Society \textbf{464}
  (2017), no.~2, 2350--2355.

\bibitem{wisdom1991symplectic}
Jack Wisdom and Matthew Holman, \emph{Symplectic maps for the {$N$}-body
  problem}, The Astronomical Journal \textbf{102} (1991), 1528--1538.

\bibitem{wisdom1996correctors}
Jack Wisdom, Matthew Holman, and Jihad Touma, \emph{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.

\bibitem{xia1992existence}
Zhihong Xia, \emph{The existence of noncollision singularities in {N}ewtonian
  systems}, Annals of Mathematics \textbf{135} (1992), no.~3, 411--468.

\bibitem{yoshida1990construction}
Haruo Yoshida, \emph{Construction of higher order symplectic integrators},
  Physics Letters A \textbf{150} (1990), no.~5-7, 262--268.

\end{thebibliography}


\end{document}