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1.Kovalchuk V., Mladenov I.M., Mechanics of infinitesimal gyroscopes on Mylar balloons and their action-angle analysis, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6099, pp.1-12, 2020
Kovalchuk V., Mladenov I.M., Mechanics of infinitesimal gyroscopes on Mylar balloons and their action-angle analysis, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6099, pp.1-12, 2020

Abstract:
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). Moreover, in both non-geodetic models, the action variables were intertwined with the quantum number N corresponding to the quantized value of the radius r of the inflated Mylar balloon.

Keywords:
action-angle analysis, affinely-rigid bodies, Bohr-Sommerfeld quantum theory, Hamilton-Jacobi equation, infinitesimal gyroscopes, Mylar balloons, residue analysis.