ИСТИНА

A problem of the motion of a heavy particle on a revolving sphere under viscous or dry friction condition is considered. In the case of viscous friction the existence, bifurcation, and stability of the equilibrium positions are discussed. A problem of existence of the periodic motions is also studied. An approach to a problem of finding them under the assumption of a small viscous friction coefficient is posed. In the case of dry friction the existence and bifurcation of the equilibria are discussed. The bifurcations of relative equilibria of a heavy bead sliding with dry friction on a circular hoop freely rotating around its vertical diameter are also considered. For this problem, steady motions of the bead are found, and their dependence on the constant of the angular momentum integral is investigated. The qualitative behaviour of the system, and in particular, the stability of the found steady motions is studied.