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://journals.pan.pl/Content/119411/PDF/26_01793_Bpast.No.69%282%29_26.04.21_K2_G_TeX_OK.pdf
L2 - http://journals.pan.pl/Content/119411
PY - 2021
IS - 2
EP - e136727
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
SP - e136727
T1 - Mechanics of infinitesimal gyroscopes on helicoid-catenoid deformation family of minimal surfaces
DA - 08.03.2021
UR - http://journals.pan.pl/dlibra/publication/edition/119411
T2 - Bulletin of the Polish Academy of Sciences: Technical Sciences
DOI - 10.24425/bpasts.2021.136727
ER -