Motion of a Lamina on a Blade under the Action of Special Type of Forces

The problem of a lamina sliding on a fixed plane under the action of special type of forces is considered. It is assumed that a blade of negligible length is fixed at some point of the plane of motion. The presence of the blade sets a nonholonomic constraint on the motion of the lamina: the velocity of the lamina point located above the blade edge must be directed along the blade. The equations of motion of the lamina are presented in the form of Gibbs–Appell equations. The obtained equations are shown to possess a manifold of equilibria. The condition that singles out the stable part of the equilibrium manifold is derived. The variable that slowly changes along the manifold of equilibria is determined, and the evolution of the slow variable in the neighborhood of the stable part of the manifold of equilibria is described (transgression).