On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D'Alembert-Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results. © 2011 Springer Science+Business Media, Inc.