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Classical motions of infinitesimal rotators on Mylar balloons
Author(s) -
Kovalchuk Vasyl,
Mladenov Ivaïlo
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6660
Subject(s) - geodesic , infinitesimal , mathematics , surface of revolution , motion (physics) , mathematical analysis , euclidean space , elliptic function , solving the geodesic equations , plane (geometry) , space (punctuation) , surface (topology) , euclidean geometry , geometry , classical mechanics , physics , linguistics , philosophy
This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two‐dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three‐dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so‐obtained results are also compared with those for the two‐dimensional sphere embedded into the three‐dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.