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

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.

Classical motions of infinitesimal rotators on Mylar balloons

“…This is true especially in connection with the considerations of these objects at the level of the old quantum theory (according to the Bohr-Sommerfeld postulates) presented in Kovalchuk and Mladenov. 6 It is worth also to mention that we have shown there that the action-angle variables for the Hamilton-Jacobi equation describing the motions of infinitesimal rotators on the Mylar balloon were intertwined with the quantum number N corresponding to the quantized value of the radius R of the inflated balloon.…”

“…Moreover, using (6) and the fact that −1 ≤ u ≤ 1, we can show that − ≥ 0. In fact, substituting the second expression in (6) into the first one with A = 1 and using the expressions (28) for the components of the first fundamental form on the Mylar balloon, we obtain that…”

“…Recently, we have also discussed in details the action-angle analysis and residue calculations for the corresponding Hamilton-Jacobi equations describing the infinitesimal rotators moving on two-dimensional surfaces with constant mean curvature (cylinders, spheres, and unduloids). 3 Here, we chose the very interesting from the geometrical point of view surface known as the Mylar balloon [4][5][6] as it is the closest "cousin" of the two-dimensional sphere in the three-dimensional Euclidean space.…”

Mechanics of incompressible test bodies moving in Riemannian spaces

“…As a continuation of the presented work, we are planning to analyse in more detail in the following papers the internal part of the motion for which the second of the solutions in (65), (67), and (68) are expressed with the help of incomplete elliptic integrals and Jacobi elliptic functions, as well as consider the motion of incompressible test bodies on different and more irregular two-dimensional surfaces embedded into the three-dimensional Euclidean space, for example, other (apart from spheres) Delaunay and minimal surfaces of constant (including zero) mean curvature (cylinders, catenoids, helicoids, unduloids, nodoids, gyroids, etc), other (apart from spheres) algebraic surfaces of the second and fourth orders (ellipsoids, pseudo-spheres, tori, etc), or quite specific but very interesting from the geometrical point of view surface which is called the Mylar balloon. 12,16,17…”

“…This work is a continuation of our recent research where we have investigated the motion of infinitesimal gyroscopes (rigid bodies) on such very interesting and instructive two-dimensional surfaces as Delaunay surfaces (spheres and cylinders as limiting cases of unduloids) of constant mean curvature 11 or Mylar balloons. 12 In the present article, we are generalizing the description to the situation of the incompressible test bodies for which apart from rotations also some deformation is allowed. Let us also describe the subject of our interest in the two-fold manner, that is, from the very beginning let us introduce some general formulation (independent of the particular form of the two-dimensional surface on which the test body is moving) and simultaneously illustrate the general procedure on the example of an incompressible test body moving on a two-dimensional spherical surface embedded into some three-dimensional Euclidean space.…”