Data Driven Modeling of a Magnetic Newton's Cradle using the Optimization of a Discrete Loss

Abstract

We investigate the dynamics of a magnetic Newton's cradle (MNC) using experiments, simulations through forward and inverse data driven models. In the MNC magnetic forces replace the instantaneous elastic collisions of the classical design. We compare point-dipole and finite-current-loop interaction models, identifying their parameters from direct measurements as well as through pendulum trajectories using the method of Optimizing a Discrete Loss (ODIL). For a synthetic three-pendulum twin, ODIL reproduces discretized trajectories to round-off accuracy and infers accurately magnet strength and damping from sparse data even when only one pendulum is observed. Model comparison shows that the point-dipole model compensates for near-field errors by underestimating the magnetic moment and produces structured residuals at each momentum transfer. The current-loop model removes this bias and, on full-precision recordings, also recovers the magnet radius. The continuous magnetic coupling reproduces the classical cradle's key behavior while enabling richer dynamics, including $N$ normal modes and a static repulsion offset. These results demonstrate that ODIL can infer physical parameters, and quantify model-form errors from sparse, noisy observations. Ultimately, this work establishes the magnetic Newton's cradle not merely as a physical curiosity, but as a rigorously quantifiable testbed for computational inverse modeling.