Mechanics of infinitesimal gyroscopes on Mylar balloons and their action-angle analysis

摘要

Here, we apply the general scheme for description of mechanics of infinitesimal bodies in Riemannian spaces to the example of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloon. The structure of partial degeneracy is investigated with the help of the corresponding Hamilton-Jacobi equation and action-angle analysis. In all situations, it was found that for any of the sixth disjoint regions in the phase space among the three action variables only two are essential for the description of our models at the level of the old quantum theory (according to the Bohr-Sommerfeld postulates).