Rigid Bodies as Accidental Seismoscopes: Exact Time Stepper Inversion and an Estimate of Ground Acceleration in the 2026 La Guaira Earthquake

Abstract

An unknown ground excitation is recovered from the measured response of a rigid body sliding in two dimensions on an inclined surface under Coulomb friction, a system with no classical solution, whose sampled trajectories come from a time stepper. The Moreau stepper is shown to admit an exact inverse, explicit, non-iterative and vectorizable, giving the ground acceleration over each step in closed form, without regularization or an assumed source model. The inverse evaluates the friction direction at the end of the step, not its beginning. Inverting the continuous equation instead is not a secondary discretization choice: the two inverses differ by an exact identity proportional to the rotation of the sliding direction within the step, bounded by twice the frictional deceleration, which is of order one g for typical interfaces. The identity is inactive in one dimension, which explains why the sliding block literature has not confronted it. Verification against a manufactured reference solution recovers it to machine precision. On the field record, the discrepancy reaches 0.10g in root mean square, which exceeds the whole propagated uncertainty of the reported peak, and it suppresses 17% of the recovered travel, behaving as a bias, not noise, so root-mean-square agreement in acceleration understates it. The method is applied to surveillance video of a delineator post that slid in the 24 June 2026 La Guaira earthquake, yielding a peak ground acceleration of 0.912 +/- 0.047g, and a lower bound of 0.47g independent of the imagery. Displaced rigid bodies can therefore provide quantitative ground-motion information where instruments are absent.