The elusive d’Alembert-Lagrange dynamics of nonholonomic systems

Abstract

While the d’Alembert-Lagrange principle has been widely used to derive equations of state for dynamical systems under holonomic (geometric) and non-integrable linear-velocity (kinematic) constraints, its application to general kinematic constraints with a general velocity and acceleration-dependence has remained elusive, mainly because there is no clear method, whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. We show how this limitation can be resolved by requiring that the states displaced by the variation are compatible with the kinematic constraints. A set of linear auxiliary conditions on the displacements is established and adjoined to the d’Alembert-Lagrange equation via Lagrange’s multipliers to yield the equations of state. As a consequence, new transpositional relations satisfied by the velocity and acceleration displacements are also established. The theory is tested for a quadratic velocity constraint and for a nonholonomic penny rolling and turning upright on an inclined plane.