% --- Section 3.3 : Geometric Synthesis: Classical Obstructions and the Normal Cone Imperative ---
\newpage
\subsection{Geometric Synthesis: Classical Obstructions and the Normal Cone Imperative}
\label{sec:arenstorf_validation}



The planar circular restricted three-body problem serves as a foundational low-dimensional archetype. Having established the canonical synodic Hamiltonian and its Jacobi invariant in~\S\ref{sec:pcr3bp_canonical}, and connected Levi-Civita's conformal map and McGehee's boundary to the Rees deformation in~\S\ref{sec:classical_regularizations}, we now synthesize how the normal cone geometry unifies these historical constructions and why it becomes strictly indispensable for the general $N$-body problem.

\subsubsection{Canonical Synodic Splitting in the Planar Prototype}
The autonomous synodic Hamiltonian partitions naturally into kinetic-Coriolis and gravitational components, $H_{\mathrm{syn}} = H_{\mathrm{kin}} + H_{\mathrm{grav}}$:
\begin{equation}
    H_{\mathrm{kin}}(\mathbf{q}, \mathbf{p}) = \frac{1}{2}\|\mathbf{p}\|^2 + (y p_x - x p_y), \qquad
    H_{\mathrm{grav}}(\mathbf{q}) = -\frac{1-\mu}{r_1} - \frac{\mu}{r_2}.
\end{equation}
The kinematic Hamiltonian $H_{\mathrm{kin}}$ generates a linear autonomous vector field on $T^*\R^2$ whose exact flow over time-step $h$ is given by the closed symplectic mapping $\Psi_h^{\mathrm{kin}} \in \operatorname{Sp}(4, \R)$:
\begin{equation}
    \mathbf{q}(h) = \mathbf{R}_{-h} \big(\mathbf{q}(0) + h \mathbf{p}(0)\big), \qquad
    \mathbf{p}(h) = \mathbf{R}_{-h} \mathbf{p}(0),
\end{equation}
where $\mathbf{R}_{-h} = \begin{pmatrix} \cos h & \sin h \\ -\sin h & \cos h \end{pmatrix} \in \mathrm{SO}(2)$ is computed without truncation error. The gravitational potential $H_{\mathrm{grav}}$ depends exclusively on positions $\mathbf{q}$, acting as an exact momentum translation:
\begin{equation}
    \mathbf{p}(h) = \mathbf{p}(0) - h \nabla_\mathbf{q} H_{\mathrm{grav}}(\mathbf{q}(0)), \qquad \mathbf{q}(h) = \mathbf{q}(0).
\end{equation}
Their symmetric composition $\mathcal{S}_2(h) = \Phi_{h/2}^{\mathrm{grav}} \circ \Psi_h^{\mathrm{kin}} \circ \Phi_{h/2}^{\mathrm{grav}}$ yields a second-order, time-reversible symplectic integrator preserving the phase space 2-form $\omega_0$ unconditionally. (The quantitative numerical validation of this normal cone scheme along the highly sensitive Arenstorf figure-eight periodic orbit~\cite{arenstorf1963periodic} ($T \approx 17.0652$, $\mu = 0.01215$), demonstrating machine-precision shadow Jacobi conservation $|\Delta C_J| < 10^{-14}$ and long-term phase stability across repeated lunar close encounters, is provided in the executable reference implementation of Appendix~\ref{app:arenstorf_solver_sec} alongside the full benchmark comparisons in Section~\ref{ch:ch07_benchmarks_and_lean4}.)
\pythonrefright{py_interfaces}{interfaces/pcr3bp_canonical_solver.py}{Planar PCR3BP Float64}

\begin{rembox}[Normal Cone Subsumption and Resolution of Classical Obstructions, see~{\cite{sundman1912memoire, levicivita1920regularisation, mcgehee1974triple, calaque2024deformation}}]
\label{rem:conormal_subsumption}\label{thm:conormal_subsumption}%
The cotangent lift of the normal cone deformation $\mathcal{D}_Z M \to \mathbb{A}^1$, formalized in the unshifted symplectic framework following Calaque--Safronov, provides a unifying geometric umbrella subsuming classical low-dimensional regularizations while resolving four fundamental structural deadlocks:
\begin{enumerate}[leftmargin=1.8em, itemsep=3pt, topsep=2pt]
    \item \textbf{Subsumption of Levi-Civita and McGehee:} In the binary collision limit $r_k \to 0$, the cotangent bundle of the normal bundle $T^*(N_{\Delta_k} M)$ fibers over the collision locus with flat abelian Lie algebra fibers. Levi-Civita's conformal scaling $|J_{LC}| = 4r$ and McGehee's scale-invariant coordinates $\rho = r^{1/2}$ emerge as specific one-dimensional coordinate slices of this Rees deformation.
    \item \textbf{Resolution of Sundman's Time-Freezing Dilemma:} Sundman's transformation $\mathrm{d}t = r\,\mathrm{d}\tau$ regularizes isolated binary collisions by slowing global time, but freezes the physical clock for all distant uncolliding bodies when $N \ge 3$. The normal cone splitting operates at constant physical time $t$, desingularizing momentum transfers along vertical cotangent fibers without time distortion.
    \item \textbf{Resolution of Non-Abelian Coordinate Branching:} Conformal coordinates---whether the 2D Levi-Civita spinor map $z = w^2$ or the 4D Kustaanheimo--Stiefel (KS) spinor transformation~\cite{kustaanheimo1965perturbation}---cannot be consistently glued across multiple overlapping collision divisors $\Delta_{ij} \cap \Delta_{jk} \ne \emptyset$. In systems with $N \ge 3$, spatial spinor transformations fail to commute along triple collision branches, resulting in non-abelian coordinate branching ambiguities and singular coordinate patches. The normal cone framework substitutes non-local coordinate transformations with a local canonical base-fiber (or Lie--Poisson) splitting on the cotangent bundle.
    \item \textbf{Resolution of McGehee's Non-Symplectic Boundary:} McGehee's collision manifold $\mathbf{M}_C$ excises collisions by introducing a manifold with boundary that loses the canonical symplectic form $\omega_0$. By contrast, the Rees algebra deformation preserves exact relative symplecticity $\omega_t$ on all fibers.
\end{enumerate}
\end{rembox}

This conceptual subsumption demonstrates that pre-existing regularizations are low-dimensional shadows of the deformation to the normal cone. To extend this geometry to arbitrary particle numbers $N \ge 4$ without coordinate singularities, one must resolve the intersecting collision arrangement $\Delta$ via the multiscale Fulton--MacPherson compactification $\mathbf{Q}[N]$, developed in Section~\ref{ch:ch04_fulton_macpherson_splitting}.