The Bouncing Penny and Nonholonomic Impacts

TL;DR

This paper extends variational principles to nonholonomic hybrid systems, deriving corner conditions and analyzing impacts in systems like a rolling disk billiard, bridging a gap in understanding nonholonomic impacts.

Contribution

It develops a variational framework for nonholonomic hybrid systems by deriving Weierstrass-Erdmann corner conditions, enabling analysis of impacts in such systems.

Findings

Derived nonholonomic corner conditions using the Lagrange-d'Alembert principle.

Applied the framework to analyze impacts in a rolling disk billiard.

Established a connection between variational impacts and nonholonomic constraints.

Abstract

The evolution of a Lagrangian mechanical system is variational. Likewise, when dealing with a hybrid Lagrangian system (a system with discontinuous impacts), the impacts can also be described by variations. These variational impacts are given by the so-called Weierstrass-Erdmann corner conditions. Therefore, hybrid Lagrangian systems can be completely understood by variational principles. Unlike typical (unconstrained / holonomic) Lagrangian systems, nonholonomically constrained Lagrangian systems are not variational. However, by using the Lagrange-d'Alembert principle, nonholonomic systems can be described as projections of variational systems. This paper works out the analogous version of the Weierstrass-Erdmann corner conditions for nonholonomic systems and examines the billiard problem with a rolling disk.