DOI 10.15507/2079-6900.23.202102.159–170

Original article

ISSN 2587-7496 (Print)

ISSN 2079-6900 (Online)

MSC2020 34A34, 34A05, 34C25, 34C27

On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces

A. A. Kosov, E. I. Semenov

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (Irkutsk, Russian Federation)

Abstract. A nonlinear system of differential equations describing the rotational motion of a rigid body under the action of torque of potential and circular-gyroscopic forces is considered. For this torque, the system of differential equations has three classical first integrals: the energy integral, the area integral, and the geometric integral. For the analogue of the Lagrange case, when two moments of inertia coincide and the potential depends on one angle, an additional first integral is found and integration in quadratures is performed. A number of examples is considered where parametric families of exact solutions are considered. In these examples, polynomial or analytical functions were used as a potential. In particular, we construct families of periodic and almost periodic motions, as well as families of asymptotically uniaxial rotations. We also identified movements that have limit values of opposite signs for unlimited increase and decrease of time.

Key Words: rigid body, equations of motion, first integrals, exact solutions

For citation: A. A. Kosov, E. I. Semenov. On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:2(2021), 159–170. DOI: https://doi.org/10.15507/2079-6900.23.202102.159–170

Submitted: 18.02.2021; Revised: 19.03.2021; Accepted: 01.05.2021

Information about the authors:

Alexander A. Kosov, Leading Researcher, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (134 Lermontov St., Irkutsk 664033, Russia), Ph. D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0003-1352-1828, kosov_idstu@mail.ru

Eduard I. Semenov, Senior Researcher, Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of Russian Academy of Sciences (134 Lermontov St., Irkutsk 664033, Russia), Ph. D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-9768-9945, edwseiz@gmail.com

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.