A Global Quasi-Symplectic Projection Method for the Spatial N-Body Problem: Explicit Preservation of Poincaré–Noether Invariants with O(N) Complexity

Abstract

Long-term numerical integration of the spatial N-body problem suffers from systematic unphysical drift of the fundamental integrals of motion: total energy E_tot, linear momentum P_tot, and angular momentum L_tot. Standard high-order explicit integrators (e.g., classical 4th-order Runge-Kutta) accumulate numerical dissipation, leading to unphysical orbital degradation and artificial nodal precession. We present a non-iterative algorithmic layer—a global quasi-symplectic projector—that forcibly returns the phase-space trajectory onto the exact invariant manifolds at the end of every time step. Formulated entirely in Cartesian vector operations with the Rodrigues rotation operator R_rot ∈ SO(3), the method eliminates coordinate singularities and retains linear computational complexity O(N). Numerical experiments on a hierarchical triple system with a highly eccentric, inclined orbit confirm that the approach suppresses secular chaos and holds all Poincaré-Noether invariants at machine precision (~10^-15) with less than 5% overhead.