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Theor. Appl. Mech., 2020, Volume 47, Issue 2, Pages 257–287 (Mi tam89)  

Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Vladimir Dragovićab, Borislav Gajića, Božidar Jovanovića

a Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia
b Department of Mathematical Sciences, University of Texas at Dallas, Dallas, USA

Abstract: We present an integrable nonholonomic case of rolling without sliding of a gyroscopic ball over a sphere. This case was introduced and studied in detail by Vasilije Demchenko in his 1923 doctoral dissertation defended at the University of Belgrade, with Anton Bilimović as the advisor. These results are absolutely unknown to modern researchers. The study is based on the C. Neumann coordinates and the Voronec principle. By using the involved technique of elliptic functions, a detailed study of motion is performed. Several special classes of trajectories are distinguished, including regular and pseudo-regular precessions. The so-called remarkable trajectories, introduced by Paul Painlevé and Anton Bilimović, are described in the present case. The historical context of the results as well as their place in contemporary mechanics are outlined.

Keywords: nonholonimic dynamics, rolling without sliding, C. Neumann coordinates, elliptic functions, elliptic integrals, Voronec principle, regular and pseudo-regular precessions, remarkable trajectories.

Funding Agency Grant Number
Ministry of Education, Science and Technical Development of Serbia
This research has been partially supported by the Mathematical Institute of the Serbian Academy of Sciences and Arts and the Ministry of Education, Science, and Technological Development of Serbia.


DOI: https://doi.org/10.2298/TAM201106015D

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Bibliographic databases:

MSC: Primary 37J60, 01A60, 33E05, 01A72; Secondary 70F25, 53Z05
Received: 06.11.2020
Revised: 30.11.2020
Language:

Citation: Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287

Citation in format AMSBIB
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\by Vladimir~Dragovi\'c, Borislav~Gaji\'c, Bo{\v z}idar~Jovanovi{\'c}
\paper Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis
\jour Theor. Appl. Mech.
\yr 2020
\vol 47
\issue 2
\pages 257--287
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\crossref{https://doi.org/10.2298/TAM201106015D}
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