Motion of a Model System Near Intersecting Curves

The paper considers the motion of a mechanical system near a geometric singularity of the configuration space of the type of two intersecting lines on a plane. This type of singularity arises in mechanical systems with holonomic constraints when the number of constraints is 1 less than the number of generalized coordinates. It is assumed that holonomic constraints become dependent at one isolated point, where the rank of constraints decreases by 1. We investigate the influence of a generalized force that is orthogonal to possible displacements on the motion of a holonomic system near a singularity of the configuration space. We prove that, for a non-degenerate singularity, the Lagrange multipliers become unbounded when the trajectory moves toward a singular point under the action of an “orthogonal” force. Because of that, the model of holonomic dynamics must be refined near singular points. To resolve the uncertainty, in this paper, we use a method in which holonomic constraints are realized as an elastic potential with a large stiffness parameter. We consider a model problem of the motion of a material point along the union of coordinate axes in the plane. When integrating numerically, it turns out that the trajectories of a system with a stiff potential can differ from the trajectory of a system with holonomic constraints. For a holonomic system, we obtain uniform rectilinear motion along one axis. Trajectories of a system with a stiff potential can periodically move away and return into a neighborhood of a singular point, switch to motion near another axis, or move for a finite time within a small neighborhood of a singular point.