Rough Collisions

TL;DR

This paper characterizes and constructs rough collision laws for rigid bodies with surface asperities, focusing on a disk and wall in 2D, revealing conditions for symmetry and velocity conservation.

Contribution

It provides a complete characterization of rough collision laws under symmetry and conservation constraints, and offers a method to explicitly construct such laws.

Findings

Collision laws are symmetric and conserve certain phase space projections.

All such laws are characterized by the invariance and conservation properties.

A method for explicit construction of rough collision laws is developed.

Abstract

A rough collision law describes the limiting contact dynamics of a pair of rough rigid bodies, as the scale of the rough features (asperities) on the surface of each body goes to zero. The class of rough collision laws is quite large and includes random elements. Our main results characterize the rough collision laws for a freely moving rough disk and a fixed rough wall in dimension 2. Any collision law which (i) is symmetric with respect to a certain well-known invariant measure from billiards theory, and (ii) conserves the projection of the phase space velocity onto the "rolling velocity" is a rough collision law. We also provide a method for explicitly constructing rough collision laws for a broad range of choices of microstructure on the disk and wall. In our introduction, we review past work in billiards, including characterizations of other rough billiard systems, which our results build upon.