On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis)

The motion of a heavy point on the surface of a rotating sphere is considered. It is assumed that the rotation axis does not coincide with the vertical diameter of the sphere and the angular velocity of the sphere is constant. The Lagrange equations for this system are derived. Sets of relative equilibria are found and their dependence on the parameters of the system is studied in extreme cases when the magnitude of the angular velocity or the distance between the rotation axis and the center of the sphere is large. The results are represented in graphic form. The same graphic series are also numerically plotted in the general case.