This article considers the motion of a two-dimensional mechanical system near a configuration space which is locally structured as two intersecting lines on a plane. The intersection point of the lines is a geometric singular point. The motion of a holonomic mechanical system near the intersection of the lines is not described by means of Lagrange’s equations of the second kind because there are no well-defined generalized coordinates. In order to analyze the system’s motion, a method is used which implements the reaction forces of holonomic constraints using elastic forces with a large stiffness parameter. Previously, this method of implementing constraints was considered for systems with a smooth configuration space, but it is also could be applied to systems with geometric singularities. This method allows one to derive equations of motion for a finite stiffness parameter. Therefore, it is possible to study the behavior of trajectories with increasing stiffness of the potential. For a singularity of two intersecting lines, it is shown that for conservative mechanical systems, a “singular square” exists in the neighborhood of the singular point. Any trajectory that intersects the neighborhood of the singular point must pass through the singular square. Equations of motion for the system on the scale of a singular square are obtained. It is shown that for motion toward a singular point along the first line, in most cases the existence of the second line could be “forgotten”, for example, if the initial velocity vector is tangent to the constraints manifold. However, if the initial velocity vector has a sufficiently large “orthogonal” component, the system’s trajectory can “hang” near the singular square. The mechanical system periodically exits and returns to the singular square, which complicates the analysis of the conditions for “exiting” from the neighborhood of the singular point.
Keywords: holonomic constraints, singular point, manifolds with singularities, Lagrange multipliers, realization of holonomic constraints
Received November 24, 2025
Revised January 16, 2026
Accepted January 19, 2026
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 two-dimensional mechanical system near a singular square. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2026, vol. 32, no. 1, pp. 61–76.
