On the Dynamics of a Rigid Body in the Hess Case at High-Frequency Vibrations of a Suspension Point

The motion of a heavy rigid body with a mass geometry corresponding to the Hess case is considered. The suspension point of the body is assumed to perform high-frequency periodic vibrations of small amplitude in the three-dimensional space. It is proved that for any law of vibrations of this type, the approximate autonomous equations of the body motion admit an invariant relation (the first integral at the zero level), which coincides with a similar relation that exists in the Hess case of the motion of a body with a fixed point. In the approximate equations of motion written in Hamiltonian form, the cyclic coordinate is introduced and the corresponding reduction is performed. For the laws of vibration of the suspension point corresponding to the integrable cases (when there is another cyclic coordinate in the system), a detailed study of the model one-degree-of-freedom system is given. For the nonintegrable cases, an analogy with the approximate problem of the motion of a Lagrange top with a vibrating suspension point is drawn, and the results obtained earlier for the top are used. Some properties of the body motion at the nonzero level of the above invariant relation are also discussed.