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1 Bulletin of the Polish Academy of Sciences. Technical Sciences

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Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces

Kovalchuk Vasyl, Gołubowska Barbara, Mladenov Ivaïlo M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2021

Vol. 69, nr 2

art. no. e136727

In this paper we explore the mechanics of infinitesimal gyroscopes (test bodies with internal degrees of freedom) moving on an arbitrary member of the helicoid-catenoid family of minimal surfaces. As the configurational spaces within this family are far from being trivial manifolds, the problem of finding the geodesic and geodetic motions presents a real challenge. We have succeeded in finding the solutions to those motions in an explicit parametric form. It is shown that in both cases the solutions can be expressed through the elliptic integrals and elliptic functions, but in the geodetic case some appropriately chosen compatibility conditions for glueing together different branches of the solution are needed. Additionally, an action-angle analysis of the corresponding Hamilton-Jacobi equations is performed for external potentials that are well-suited to the geometry of the problem under consideration. As a result, five different sets of conditions between the three action variables and the total energy of the infinitesimal gyroscopes are obtained.