Complete inequivalence of nonholonomic and vakonomic mechanics

Nonholonomic mechanics, which relies on the method of Lagrange multipliers in the d'Alembert-Lagrange formulation of the classical equations of motion, is the standard description of the dynamics of systems subject to nonholonomic constraints. Vakonomic mechanics has been proposed as another possible approach to the dynamics of nonholonomic systems. Partial equivalence theorems have been found that provide conditions under which nonholonomic motions are also vakonomic motions. The aim of the present work is to show that these results do not cover at least one important physical system, whose motion brought about by vakonomic mechanics is completely inequivalent to the one derived from nonholonomic mechanics. 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.