TY - JOUR N2 - 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. L1 - http://www.czasopisma.pan.pl/Content/119411/PDF/26_01793_Bpast.No.69(2)_26.04.21_K2_G_TeX_OK.pdf L2 - http://www.czasopisma.pan.pl/Content/119411 PY - 2021 IS - 2 EP - e136727 DO - 10.24425/bpasts.2021.136727 KW - action-angle analysis KW - mechanics of infinitesimal gyroscopes KW - geodesic and geodetic equations of motion KW - helicoid-catenoid deformation family of minimal surfaces KW - elliptic integrals and elliptic functions A1 - Kovalchuk, Vasyl A1 - Gołubowska, Barbara A1 - Mladenov, Ivaïlo M. VL - 69 DA - 08.03.2021 T1 - Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces SP - e136727 UR - http://www.czasopisma.pan.pl/dlibra/publication/edition/119411 T2 - Bulletin of the Polish Academy of Sciences Technical Sciences ER -