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

TL;DR

This paper revisits Demchenko's 1923 integrable nonholonomic model of a gyroscopic ball rolling over a sphere, revealing previously unknown results and analyzing special trajectories using elliptic functions.

Contribution

It uncovers and details Demchenko's original integrable case, applying modern techniques to analyze its trajectories and historical significance.

Findings

Identification of integrable nonholonomic case from 1923

Description of special trajectories including precessions

Application of elliptic functions to analyze motion

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 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. So-called remarkable trajectories, introduced by Paul Painleve and Anton Bilimovic, are described in the present case. The historic context as well as the place of the results in contemporary mechanics are outlined.