On the Orbital Stability of Pendulum Motions of a Rigid Body in the Hess Case

Given a heavy rigid body with one fixed point, we investigate the problem of orbital stability of its periodic motions. Based on the analysis of the linearized system of equations of perturbed motion, the orbital instability of the pendulum rotations is proved. In the case of pendulum oscillations, a transcendental situation occurs, when the question of stability cannot be solved using terms of an arbitrarily high order in the expansion of the Hamiltonian of the equations of perturbed motion. It is proved that the pendulum oscillations are orbitally unstable for most values of the parameters.