Topological atlas of the Kowalevski--Yehia gyrostat: analytical results and topological analysis

TL;DR

This paper reviews fifty years of research on the integrable motion of a heavy gyrostat of Kowalevski type, providing rigorous proofs and correcting previous inaccuracies in its topological analysis.

Contribution

It offers a comprehensive and rigorous topological analysis of the Kowalevski--Yehia gyrostat problem, fixing past errors and consolidating known results.

Findings

Confirmed the complete integrability of the problem.

Provided rigorous proofs of qualitative analysis.

Corrected previous inaccuracies in topological results.

Abstract

We present a review of the results obtained during the last fifty years in the problem of the motion of a heavy gyrostat under the conditions of the Kowalevski type. Hamad M. Yehia in 1986 has proved that the problem is complete integrable. Since then, a lot of works were devoted to different aspects of integrating this problem and of its topological investigation. The main idea of this work is to prove strictly all the facts of qualitative analysis and to fix different mistakes and inaccuracies. The work was supported by the grants of the RFBR No. 10-01-00043, 10-01-97001, 13-01-97025, and 14-01-00119.