Nonholonomic billiards and bounded motion in cylinders

TL;DR

The paper investigates the dynamics of a ball under gravity inside cylinders with no-slip contact, revealing bounded vertical motion under certain initial conditions. It introduces nonholonomic billiards—a smooth rolling model in one higher dimension—that approximate no-slip billiards in the small-radius limit, establishing a precise link via the rolling collision map and energy-preserving dynamics. Through analytical results and numerical illustrations, it shows that nonholonomic billiards can illuminate the complex, impulse-driven behavior of no-slip systems, including bounded trajectories and caustic collapse in cross-sections. Finally, it extends the framework to 4D rolling on 3D cylinders, deriving a hierarchical system of equations on pancake cross-sections and arguing for a program to analyze no-slip dynamics through nonholonomic models, with implications for understanding bounded motion in cylindrical geometries.

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.

Figures (19)

• Figure 1: No-slip disc bouncing between two parallel vertical lines, under gravity, with increasing values of the acceleration-due-to-gravity parameter $g$ increasing from left to right. On the far left, $g=0$ and the trajectory is periodic of period $2$.
• Figure 2: Left: segment of trajectory of the center of a ball rolling on the inner surface of a cylinder, under gravity, showing harmonic vertical motion. Center: trajectory of a no-slip billiard ball, with very short bouncing steps, not satisfying the initial rolling impact assumption. Right: under that assumption, the no-slip billiard ball closely tracks the rolling motion.
• Figure 3: Height of the ball's center as a function of time when the rolling first impact assumption is not satisfied. The sequence of graphs (NW $\rightarrow$ NE $\rightarrow$ SW $\rightarrow$ SE) was generated with decreasing lengths of the skipping motion steps. An apparent energy dissipation into the zig-zag motion at smaller and smaller scales produces accelerated falling and decreasing amplitude of oscillation.
• Figure 4: Left: a two-dimensional no-slip billiard system on a Sinai billiard table. Right: the associated $3$-dimensional rolling system. With appropriately adjusted moment of inertia parameters for the disc and the ball, the rolling-around-the-edge of the circular hole, on the right, converges to the no-slip billiard reflection off the circular scatterer, on the left, as the radius of the ball approaches $0$. This suggests that the two systems have comparable dynamical properties, and that the rolling system, governed by differential equations, can help us to understand the other----impact driven----system. The one on the left may be regarded as the two-dimensional shadow of that on the right, while the one on the right defines a soft-collisions version of the other.
• Figure 5: Trajectory of a no-slip billiard system on a disc. Left: trajectory of the moving disc's center, showing the characteristic double caustic. Right: $3$-dimensional configuration space trajectory, including the moving disc's rotation angle.

Theorems & Definitions (18)

• Theorem 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• Proposition 6
• proof