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

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 Bilimovic 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 Bilimovic, are described in the present case. The historical context of the results as well as their place in contemporary mechanics are outlined.

Spherical and planar ball bearings -- nonholonomic systems with invariant measures

“…We will study integrability in the spherical ball bearing problems in a separate paper. Also, it would be interesting to study the appropriate nonholonomic systems in arbitrary dimension R m , m > 3 (e.g., see [12,14,16,17,19]), or the systems where the homogeneous balls B i are replaced by the systems of the form (ball + gyroscope), which satisfy the Zhukovskii conditions (see [10,22]). 6.…”