Reaction Forces and Friction Forces in the Dynamics of Systems with Geometric Singularities

The specific features of the motion of holonomic mechanical systems with parameters are discussed. For some (critical) parameter values, the configuration space of a mechanical system is a manifold with singularities. For other parameter values, the configuration space is a smooth manifold. It is assumed that the sliding friction force according to the Amontons–Coulomb model can act upon one of the material points of the mechanical system. When the parameters of a mechanical system differ from critical, the classical Lagrange equations could be applied to describe its dynamics. The main interest is motion on smooth manifolds near points which transform into singular points as the parameters of the mechanical system tend to critical values. The behavior of reaction forces and Lagrange multipliers for such “presingular” points is considered. Two types of configuration spaces with singularities are studied: the union of two intersecting curves in the plane and the union of two tangent curves in the plane. For the first time, various behaviors of the Lagrange multipliers are described using the example of a given type of perturbation of configuration spaces with singularities. In general, it is proven that for a singularity of the intersection type, the Lagrange multipliers become unbounded near the singular point (on a manifold with singularities), regardless of the influence of the friction force. For a tangency singularity, there are different variants taking into account the friction force. For one type of perturbation of the configuration space, the resulting Lagrange multipliers are bounded. For another type of perturbation of the configuration space, the resulting Lagrange multipliers are unbounded. The general property of the friction force for the considered mechanical systems is derived. If the friction force is taken into account, then there are two solutions for reaction forces when moving near a singular point in one direction, but there are no solutions when moving in the other direction.