math.DS — Dynamical Systems

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

Contributed by Jeremy Nusa

In the classical formulation of the Newtonian $N$-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 $T^*\mathbf{Q}[N]$ 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 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 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.