doiSerbia

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

Dragović Vladimir (Department of Mathematical Sciences, University of Texas at Dallas, Dallas, USA + Mathematical Institute Serbian Academy of Sciences and Arts Belgrade Serbia), Vladimir.Dragovic@utdallas.edu
Gajić Borislav (Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia), gajab@mi.sanu.ac.rs
Jovanović Božidar (Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia), bozaj@mi.sanu.ac.rs

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 pseudoregular precessions. The so-called remarkable trajectories, introduced by Paul Painlev’e 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