New feature in hoop dynamics: hidden jump

Dynamics of nonhomogeneous disk on a rough support is considered. Since P. Painlevé, this system becomes a popular simple model to demonstrate singularities intrinsic to multibody dynamics. It is shown that equations of motion can be ambiguous: along with continued rolling the body can jump or slide with nonzero initial acceleration. An extended model of the system is applied to stability analysis and exhibits bistability, where rolling and jump correspond to sinks, while sliding turns to be a saddle point. In previous studies, the support was treated as a horizontal plane, and such paradox was not detected. Here the cases of an inclined plane and a half-pipe are examined. In particular, such periodic motions with possible jumps are constructed, where continuous rolling is possible too. Generally speaking, “hidden” dynamics is an essential feature of multibody systems, not taken in account in straightforward computing.