An approach in studying gyrostat motion with variable gyrostatic moment

The problem of the motion of a gyrostat with a fixed point and a variable gyrostatic moment under the action of gravity force is considered. A new method for integrating the equations of motion of a system consisting of a carrier body and three rotors that rotate around the main axes is proposed. The method can be attributed to the method of variation of the constant in the function for the gyrostatic moment, which linearly depends on the vector of vertical. In case of a constant multiplier, the gyrostatic moment satisfies the Poisson equation, and its variation is found from the integral of areas. The original equations have been reduced to a fifth-order system. New solutions of these equations are obtained in the case of a spherical mass distribution for the gyrostat and for the precessional motions of a carrier body. An explicit form of the gyrostatic moment is established for the case of three invariant relations.