Classical motions of infinitesimal rotators on Mylar balloons |
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Authors: | Vasyl Kovalchuk Ivaïlo Mladenov |
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Affiliation: | 1. Department of Theory of Continuous Media and Nanostructures, Institute of Fundamental Technological Research, Polish Academy of Sciences, 5B, Pawińskiego Str., Warsaw, 02-106 Poland;2. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 21, Sofia, 1113 Bulgaria |
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Abstract: | 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. |
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Keywords: | elliptic integrals and elliptic functions geodesics and geodetics infinitesimal rotators Mylar balloons non-Euclidean spaces surfaces of revolution |
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