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

Vakonomic mechanics has been proposed as a possible description of the\ndynamics of systems subject to nonholonomic constraints. The aim of the present\nwork is to show that for an important physical system the motion brought about\nby vakonomic mechanics is completely inequivalent to the one derived from\nnonholonomic mechanics, which relies on the standard method of Lagrange\nmultipliers in the d'Alembert-Lagrange formulation of the classical equations\nof motion. For the rolling coin on an inclined plane, it is proved that no\nnontrivial solution to the equations of motion of nonholonomic mechanics can be\nobtained in the framework of vakonomic mechanics. This completes previous\ninvestigations that managed to show only that, for certain mechanical systems,\nsome but not necessarily all nonholonomic motions are beyond the reach of\nvakonomic mechanics. Furthermore, it is argued that a simple qualitative\nexperiment that anyone can perform at home supports the predictions of\nnonholonomic mechanics.\n