On the selection of a proper connection in describing the dynamics of constrained mechanical systems

This study is focused on a critical issue related to the direct and consistent application of Newton's law of motion to a special but large class of mechanical systems, involving equality motion constraints. For these systems, it is advantageous to employ the general analytical dynamics framework, where their motion is represented by a curve on a non-flat configuration manifold. The geometric properties of this manifold, providing the information needed for setting up the equations of motion of the system examined, are fully determined by two mathematical entities. The first of them is the metric tensor, whose components at each point of the manifold are obtained by considering the kinetic energy of the system. The second geometric entity is known as the connection of the manifold. In dynamics, the components of the connection are established by using the set of the motion constraints imposed on the original system and provide the torsion and curvature properties of the manifold. Despite its critical role in the dynamics of constrained systems, the significance of the connection has not been investigated yet at a sufficient level in the current engineering literature. The main objective of the present work is to first provide a systematic way for selecting this geometric entity and then illustrate its role and importance in describing the dynamics of constrained systems.