Abstract:
The motion of a material point near a singularity of the type of two tangent surfaces is considered. The surfaces are located symmetrically with respect to a common tangent plane and have a common axis of rotation. First, a model of motion for holonomic mechanics is considered. It is shown that only trajectories in a fixed plane containing the axis of rotation of the surfaces can pass through a singular point. At the point of contact, dynamic uncertainty arises, since the trajectory has several possible branches of motion. To study the motion of a material point near a singularity of the type of double tangent paraboloid, a model of the implementation of holonomic constraints through an elastic potential with a large stiffness parameter, or a “stiff potential”, is considered. The potential must vanish on the manifold with singularities and must be strictly positive outside it. For a model with a stiff potential, it also turns out that only trajectories in a fixed plane containing the axis of rotation of the paraboloids can pass through a singular point. Numerical modeling of the dynamics was done. It was found that the trajectories of a system with a stiff potential can qualitatively differ from the trajectories of the corresponding holonomic system. A holonomic system instantly passes a geometric singularity, moving with a non-zero velocity. A system with a stiff potential can move in a singular region for a finite time, resulting in rapid changes in the direction of the velocity vector. In real mechanical systems, this type of motion can lead to breakdowns or instability of trajectories.
Key words:singular point, manifolds with singularities, tangency type singularity, holonomic constraints, realization of constraints in mechanics.
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