On the Movement of Gyrostat under the Action of Potential and Gyroscopic Forces

A system of differential equations is considered that describes the motion of a gyrostat under the action of the moment of potential, gyroscopic and circular-gyroscopic forces. The form of the moment of forces is indicated for which the system has the three first integrals of a given form. An analog of V.I. Zubov's theorem for representing solutions of gyrostat equations by power series is given, and the possibility of using this approach to predict motions is shown. For an analogue of the Lagrange case, integration in quadratures is performed. Analogues of the case of full dynamical symmetry and the Hess case are also indicated. Based on the principle of optimal damping developed by V.I. Zubov, a design of the control moment created by circular-gyroscopic forces is proposed, which ensures that one of the coordinates reaches a constant (albeit unknown in advance) value or the transition of the state vector to the level surface of the particular Hess integral. A numerical example is given, for which a two-parameter family of exact almost periodic solutions, represented by trigonometric functions, is found.