Dynamics of the no-slip Galton board

TL;DR

This paper investigates the complex dynamics of a no-slip Galton board model, revealing how external forces influence regularity, periodicity, and the emergence of new invariant structures through numerical and analytical methods.

Contribution

It introduces a more realistic no-slip collision model for the Galton board, analyzing how external forces affect its dynamical behavior and structure, which was not previously studied.

Findings

Regularity persists under small forces.

Stronger forces create invariant regions and new structures.

External forces induce proliferating periodicity and novel orbits.

Abstract

The ideal Galton board and Lorentz gas billiard models have been studied numerically and analytically primarily in settings where friction and rotational velocity are neglected. We eliminate these simplifying assumptions and study the resulting dynamics of a more general model using no-slip collisions, in which particles rotate and may exchange linear and angular momentum at collisions while adhering to certain conservation laws. Using numerical experiments and phase portrait analysis we show that (in contrast to specular dispersing billiards) regularity persists when a small force is introduced while (consistent with specular billiards) under a stronger force new structure including invariant regions may arise. We also show analytically that with the introduction of an external force periodicity proliferates, with new types of periodic orbits not present in the no-force case.