Analytical Constraint Reconstruction for Sundman-Regularized Explicit Integration of the Kepler Problem

Abstract

We introduce Analytical Constraint Reconstruction (ACR), a closed-form, non-iterative constraint-reconstruction procedure for explicit numerical integration of the Kepler problem. The method exploits the polar decomposition of velocity into radial and tangential components. Given a prescribed angular-momentum magnitude, the tangential velocity is reconstructed directly from the current radius and mass, after which the radial velocity is obtained from the prescribed energy using the radial-motion sign from the numerical state. The contribution is deliberately narrower than a claim of inventing invariant preservation or manifold projection. Existing work has established Kepler manifold correction, explicit scaling and transformation procedures, and analytical Kepler-solver corrections. ACR instead formulates the simultaneous energy and angular-momentum reconstruction as a sequential closed-form operation in the instantaneous radial-tangential frame. For a fixed-dimensional state it requires zero nonlinear iterations, Jacobian construction, or matrix inversion and has constant O(1) reconstruction work per timestep. ACR is combined with Sundman time regularization, which addresses temporal resolution near periapsis. The existing benchmark at e = 0.90, a = 1 over 50 orbital periods reports approximately machine-precision preservation of the selected invariants for the dual reconstruction schemes. The present manuscript explicitly distinguishes constraint satisfaction from trajectory and long-term phase accuracy. A four-method equal-NRHS factorial benchmark across three eccentricities has been executed and is reported in full.