Nonholonomic billiards and bounded motion in cylinders

TL;DR

This paper introduces nonholonomic billiards as a smooth approximation to no-slip billiards, enabling better analytical understanding of complex bouncing dynamics of elastic balls in cylinders, including bounded motion under gravity.

Contribution

It proposes a novel comparison approach between no-slip and nonholonomic billiards, providing insights into their dynamics and demonstrating the utility of the smooth approximation through numerical studies.

Findings

Nonholonomic billiards effectively approximate no-slip billiards.

Bounded axial motion observed in certain gravitational conditions.

Numerical case studies illustrate the approach's usefulness.

Abstract

A widely used mathematical model for the bouncing motion of an ideally elastic ball -- referred to in previous work by the first two authors and collaborators as a {\em no-slip billiard} system -- exhibits some notable dynamical behavior that is not well-understood. For example, under certain initial conditions, the axial component of the position of the center of the ball moving inside a vertical solid cylinder under constant gravitational force does not accelerate downward as might be expected but remains bounded. There is not as yet, as far as we know, any analytical study of the bouncing ball dynamics, under gravity, in general cylinders (not necessarily having a circular cross-section) in $\mathbb{R}^3$. In this paper, we propose an approach by comparing the no-slip system with a smooth approximation of it that we call {\em nonholonomic billiards}. It consists of a $4$-dimensional ball rolling on the solid $3$-dimensional cylinder. We first review earlier work on no-slip billiards and their connection with nonholonomic (rolling) systems, explain how nonholonomic billiards approximate the no-slip kind (after work by the first two authors and B. Zhao), and illustrate the relationship with a few numerical case studies that demonstrate the utility of the soft (nonholonomic) system as a helpful tool for exploring the dynamics of no-slip billiard systems.