The d’Alembert–Lagrange equation exploited on a Riemannian manifold

In this paper, we present a geometric exploitation of the d’Alembert–Lagrange equation (or alternatively, Lagrange form of the d’Alembert’s principle) on a Riemannian manifold. We develop the d’Alembert–Lagrange equation in a geometric form, as well as an explicit analytic form with respect to an arbitrary frame in a coordinate neighborhood on the configuration manifold. We provide a procedure to determine the governing dynamic equations of motion. Examples are given to illustrate the new formulation of dynamic equations and their relations to alternative ones. The objective is to provide a generalized perspective of governing equations of motion and its suitability for studying complex dynamic systems subject to nonholonomic constraints.