Mechanics of the Infinitesimal Gyroscopes on the Mylar Balloons and Their Action-Angle Analysis

TL;DR

This paper studies the mechanics of infinitesimal gyroscopes on Mylar balloons using Hamilton-Jacobi and action-angle methods, revealing how phase space regions relate to quantum numbers and model potentials.

Contribution

It applies a general Riemannian mechanics scheme to specific models of infinitesimal rotators on Mylar balloons, analyzing degeneracy and quantization effects.

Findings

Only two of three action variables are essential in phase space.

Action variables are linked to the quantum number N for balloon radii.

The study distinguishes geodetic and non-geodetic motion models.

Abstract

Here we apply the general scheme for description of the mechanics of infinitesimal bodies in the Riemannian spaces to the examples of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloons. 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 six disjoint regions in the phase space among the three action variables only two of them 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 quantization of the radii $r$ of the inflated Mylar balloons.