Mechanics of holonomic systems with geometric singular points

In classical mechanics, configuration spaces of mechanical systems are smooth manifolds. Motion equations could be interpreted as a vector fields on the (co)tangent bundle of manifold. In the case that there are singular points in the configuration space, only particular methods are applied. Generalizations of these methods to the common theory are still far from complete. We consider several theories of singular space geometries which generalize base terms of differential calculus: (co)tangent vector, (co)tangent space and vector field, integral curves of vector fields, etc. In order to study the application of these theories to the problems of analytical dynamics, some particular examples of mechanical systems with singularities are considered. We compare kinematics and dynamics which are derived from geometric theories with observes dynamics of mechanisms’ models. This comparison could help up to formulate the conditions which generalized geometric theory of motion must satisfy. In addition, reaction forces for some perturbations of holonomic constraints near to the singularities are studied.