The application of methods for implementing holonomic constraints for describing the motion of a singular pendulum is considered. The configuration space of a singular pendulum can be represented as a union of two intersecting or two tangent smooth curves on a plane. According to the principle of constraint release, the motion of a constrained system is ensured by reaction forces that are orthogonal to virtual displacements. Classical Lagrange equations do not describe the motion of a mechanical system near geometric singularities such as branch points. In the constraint implementation method, implicitly defined reaction forces are replaced by active elastic forces with a large stiffness parameter. In publications, the issues of holonomic constraint implementation were considered only for mechanical systems with smooth configuration spaces. But the constraint implementation method naturally generalizes to systems with geometric singularities. In this work, in order to analyze the dynamics of a singular pendulum, one of its links is replaced by an extensible rod. Numerical simulation is performed for the resulting mechanical system. Several different simulation scenarios for a singular pendulum with an extensible rod are considered: motion of the system with an initial position outside the singularity, motion from an initial position at a branch point, and motion under the action of a force orthogonal to virtual displacements. The obtained trajectories agree with the smooth motion of a singular pendulum only in the first scenario. In the two other scenarios, the trajectories may differ qualitatively. Based on the observed motion trajectories, new problems for the dynamics of systems with geometric singularities in the configuration space are formulated.
S. N. Burian (Sun,) studied this question.