This paper presents the study of the dynamics of axial gyrostats. The gyrostat is composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor aligned with a principal axis of the platform. The paper discusses three types of gyrostats: oblate, prolate and intermediate. Rotation of the rotor relative to the platform provides a source of small internal angular momentum, and does not affect the moment of inertia tensor of the gyrostat. We consider the dynamics of gyrostats in the absence external torque. The dynamics are described by ordinary differential equations in the Andoyer - Deprit canonical variables. For the undisturbed motion, when the internal moment is equal to zero, the stationary solutions are found and studied their stability. Also we obtain general exact analytical solutions in terms of elliptic functions and the separatrix trajectories. These results can be interpreted as the development of the classical Euler case for a solid, when added to one degree of freedom- the relative rotation of bodies. For the disturbed motion gyrostats, when there is the system with slowly varying parameter, we obtain the adiabatic invariants in terms of complete elliptic integrals, which are approximately the first integrals of the disturbed system. The adiabatic invariants are value along a trajectory remains approximately constant on long time intervals on which the parameter changes considerably. Results of the study can be useful for the analysis of dynamics of dual-spin spacecraft and for studying the chaotic behavior of the spacecrafts.
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