Geodesic
On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 2, pp. 159-170.

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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.
Keywords: rigid body, first integrals
Mots-clés : equations of motion, exact solutions. 
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A. A. Kosov; È. I. Semenov. On exact solutions of equations of rotational motion of a rigid body under action of torque of circular-gyroscopic forces. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 2, pp. 159-170. http://geodesic.mathdoc.fr/item/SVMO_2021_23_2_a1/

[1] V. V. Golubev, Lectures on the integration of the equations of motion of a rigid body about a fixed point, Israeli Program for Scientific Translations, Israel, 1960 | MR | MR | Zbl

[2] V. I. Zubov, Analytical dynamics of the system of bodies, LSU Publ., Leningrad, 1983, 344 pp. (In Russ.)

[3] V. V. Kozlov, Qualitative analysis methods in the dynamics of a rigid body, NIC «Regulyarnaya i haoticheskaya dinamika» Publ., Izhevsk, 2000, 256 pp. (In Russ.) | MR

[4] A. V. Borisov, I. S. Mamaev, Solid dynamics, NIC «Regulyarnaya i haoticheskaya dinamika» Publ., Izhevsk, 2001, 384 pp. (In Russ.) | MR

[5] I. N. Gashenenko, G. V. Gorr, A. M. Kovalev, Classical problems in the dynamics of rigid body, Naukova dumka Publ., Kiev, 2012, 401 pp. (In Russ.) | MR

[6] S. Nikolov, N. Nedkova, “Dynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review”, Journal of Applied and Computational Mechanics, 4:1 (2015), 187–206 | DOI

[7] V. V. Kozlov, “To the problem of the rotation of a solid in a magnetic field”, Izv. AN SSSR. MTT, 6 (1985), 28–33 (In Russ.)

[8] A. V. Borisov, I. S. Mamaev, “Hess case in rigid body dynamics”, Prikladnaya matematika i mekhanika, 67:2 (2003), 256–265 (In Russ.) | MR | Zbl

[9] A. V. Belyaev, “On the general solution of the problem of the motion of a heavy rigid body in the Hess case”, Sbornik: Mathematics., 206:5 (2015), 621–649 | DOI | MR | Zbl

[10] I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess-Appelrot system and its nonholonomic analogues”, Proceedings of the Steklov Institute of Mathematics, 294:1 (2016), 252–275 | DOI | DOI | MR | MR | Zbl

[11] M. A. Novikov, “O statsionarnykh dvizheniyakh tverdogo tela pri suschestvovanii chastnogo integrala Gessa”, Izv. RAN. MTT, 3 (2018), 28–37 (In Russ.) | DOI