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
Received March 1, 2025
Revised April 21, 2025
Accepted April 28, 2025
Sergey Nikolaevich Burian, Cand. Sci (Phys.-Math.), State Research Institute of Applied Problems, St. Petersburg, 191167 Russia, e-mail: burianserg@yandex.ru
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Cite this article as: S.N. Burian. Motion of a model system near intersecting curves. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 2, pp. 38–54.
