Complete inequivalence of nonholonomic and vakonomic mechanics: rolling coin on an inclined plane

TL;DR

This paper demonstrates that for a rolling coin on an inclined plane, vakonomic mechanics cannot replicate the motions predicted by traditional nonholonomic mechanics, highlighting fundamental differences between the two approaches.

Contribution

It proves the complete inequivalence of vakonomic and nonholonomic mechanics for a specific physical system, the rolling coin on an inclined plane.

Findings

Vakonomic mechanics cannot produce nonholonomic solutions for the rolling coin.

Nonholonomic and vakonomic mechanics are fundamentally incompatible for this system.

Experimental evidence supports the predictions of nonholonomic mechanics.

Abstract

Vakonomic mechanics has been proposed as a possible description of the dynamics of systems subject to nonholonomic constraints. The aim of the present work is to show that for an important physical system the motion brought about by vakonomic mechanics is completely inequivalent to the one derived from nonholonomic mechanics, which relies on the standard method of Lagrange multipliers in the d'Alembert-Lagrange formulation of the classical equations of motion. For the rolling coin on an inclined plane, it is proved that no nontrivial solution to the equations of motion of nonholonomic mechanics can be obtained in the framework of vakonomic mechanics. This completes previous investigations that managed to show only that, for certain mechanical systems, some but not necessarily all nonholonomic motions are beyond the reach of vakonomic mechanics. Furthermore, it is argued that a simple qualitative experiment that anyone can perform at home supports the predictions of nonholonomic mechanics.