Geometrical model for a particle on a rough inclined surface

TL;DR

This paper introduces a simple geometrical model for a particle moving on a rough inclined surface, capturing various dynamical behaviors and aligning qualitatively with experimental observations.

Contribution

It presents a one-dimensional map-based model that describes the particle's motion and explores its dynamical regimes, including chaos and steady states.

Findings

Model exhibits steady, chaotic, and accelerated regimes.

Phase diagram maps dynamical regimes.

Qualitative agreement with experiments.

Abstract

A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which the dynamics is described by a one-dimensional map. This map is studied in detail and it is shown to exhibit several dynamical regimes (steady state, chaotic behavior, and accelerated motion) as the model parameters vary. A phase diagram showing the corresponding domain of existence for these regimes is presented. The model is also found to be in good qualitative agreement with recent experiments on a ball moving on a rough inclined line.