The Bouncing Penny and Nonholonomic Impacts

William Clark; Anthony Bloch

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arXiv · 1909.11192

The Bouncing Penny and Nonholonomic Impacts

William Clark·Anthony Bloch

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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.

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William Clark, Anthony Bloch. 2019-09-24. The Bouncing Penny and Nonholonomic Impacts. https://arxiv.org/abs/1909.11192

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