On the Singularities in the Dynamics of a Spherical Top with Nonholonomic Constraints

Abstract

The problem of the motion of a dynamically symmetric ball with a shifted center of gravity on a horizontal absolutely rough plane is considered, taking into account the unilateral nature of the contact. It is shown that for some values of the parameters and initial conditions, singularities of two types may appear: ambiguity in determining the subsequent motion (separation of the body from the support or continuation of rolling) or the impossibility of determining it within the framework of the model used (the finite time paradox). For a formal description of these situations, the generalized complementarity problem is used. The Littlewood problem of the rolling of a hoop with a point mass is studied in detail. Some cases of paradoxes when imposing differential constraints of various types are also discussed, including the generalization of the Chaplygin sleigh and the “rubber” ball.