Abstract:
We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that the geometry of the mass of the body corresponds to the Bobylev–Steklov case. Unperturbed motion represents oscillations or rotations of the body around a principal axis, occupying a fixed horizontal position. The problem of the orbital stability is considered on the basis of a nonlinear analysis.
In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities we study the problem analytically. In the general case we reduce the problem to the stability study of a fixed point of the symplectic map generated by equations of perturbed motion. We calculate coefficients of the symplectic map numerically. By analyzing the abovementioned coefficients we establish the orbital stability or instability of the unperturbed motion. The results of the study are represented in the form of a stability diagram.
Keywords:Hamiltonian system, periodic orbits, normal form, resonance, action-angel variables, KAM theory.
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