Hidden Harmonic Structure, Universal Damping, and Stability Bounds in Nonlinear Contact Dynamics

Abstract

Nonlinear contact dynamics are widely regarded as intrinsically nonlinear systems whose behaviour depends strongly on geometry and impact conditions. Here we show that any one-dimensional conservative contact system satisfying monotone energy-consistent conditions admits two complementary structures: (i) a canonical action-angle representation in physical time, and (ii) an exact harmonic oscillator representation under an energy-based coordinate transformation combined with time reparametrisation. This reveals a hidden linear structure underlying nonlinear contact interactions. Building on this result, we derive a unique universal damping law that preserves linear dissipative dynamics in the transformed harmonic space, and establish a rigorous, closed-form lower bound for the critical timestep in numerical simulations. The framework generalises classical power-law contact models and provides a unified basis for restitution control across arbitrary geometries, recovering known exact solutions as explicit monomial special cases.