Abstract:
The paper considers the motion of a mechanical system near a geometric singularity of the configuration space such as 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. The influence of a generalized force orthogonal to possible displacements on the motion of a holonomic system near a singularity of the configuration space is investigated. It is proven 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. Therefore, the model of holonomic dynamics must be refined near singular points. To resolve the uncertainty, this paper uses a method in which holonomic constraints are realized as an elastic potential with a large stiffness parameter. A model problem of the motion of a material point along the union of coordinate axes on a plane is considered. Through numerical integration, it has been determined that the trajectories of a system with a rigid potential can differ from the trajectory of a system with holonomic constraints. For a holonomic system, uniform rectilinear motion along one axis is obtained. The trajectories of a system with a rigid potential can periodically move away and return to the neighborhood of a singular point, switch to motion near another axis, or move for a finite time in a small neighborhood of a singular point.
Keywords:holonomic constraints, singular point, manifolds with singularities, Lagrange multipliers, realization of holonomic constraints.
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