Poisson variations in the problem of the stability of equilibria in rigid body mechanics

The existence and stability of equilibria in rigid body mechanics is considered. A class of variations is indicated which satisfy the analogue of Poisson's equations, suitable for use when investigating both the sufficient and necessary conditions for the stability of such equilibria and which, in particular, enable the invariance of the equations of motion of the system and their first integrals to be effectively used when the phase variables and parameters of the problem are interchanged. The result is illustrated using the example of the problem of the motion of a gyrostat far from attracting objects.