Integrable cases in the dynamics of axial gyrostats and adiabatic invariants

This paper presents the study of axial gyrostats dynamics. The gyrostat is composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor aligned with the platform principal axis. 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. The dynamics of gyrostats without external torque is considered. The dynamics is described by using ordinary differential equations with Andoyer–Deprit canonical variables. For undisturbed motion, when the internal moment is equal to zero, the stationary solutions are found, and their stability is studied. General analytical solutions in terms of elliptic functions are also obtained. These results can be interpreted as the development of the classical Euler case for a solid, when to one degree of freedom—the relative rotation of bodies—is added. For disturbed motion of the gyrostats, when there is a system with slowly varying parameters, the adiabatic invariants are obtained in terms of complete elliptic integrals, which are approximately equal to the first integrals of the disturbed system. The adiabatic invariants remain approximately constant along a trajectory for long time intervals during which the parameter changes considerably. The results of the study can be useful for the analysis of dynamics of dual-spin spacecraft and for studying a chaotic behavior of the spacecraft.