Teaching the Fixed Spinning Top Using Four Alternative Formulations

This paper discusses four different approaches that can be followed to derive the equations of motion for a fixed and symmetrical spinning top. Starting from the usual Euler equations in the body-fixed system, after manipulation it is shown that identical equations are derived for the space-fixe system as well. All the three Cartesian components of the angular momentum vector are calculated for both the body- and the space-systems and they are formulated so that they can be used for further numerical analysis. In addition to the classical set, the Euler equations are also easily derived using a rotating system originated at the pivot but not spinning. Moreover, Lagrange equations are derived and the latter are proven to be equivalent with the Euler equations. The best way among these four methods for teaching students is probably the instructor’s preference. Moreover, using commercial software, an adequately accurate numerical solution is derived. Not only the position of the spinning top is calculated but also the support forces at the pivot are predicted