On orbital stability pendulum-like oscillations and rotation of symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a heavy rigid body with a fixed point. We suppose that the mass center of the body is located in the equatorial plane of the inertia ellipsoid. Unperturbed motions represent oscillations or rotations of the body around a principal axis, keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of perturbed motion are obtained in Hamiltonian form. Domains of orbital instability are established by means of linear analysis. Outside of the above domains nonlinear study is performed. The nonlinear stability problem is reduced to a stability problem of a fixed point of symplectic map generated by the equations of perturbed motion. Coefficients of the above map are obtained numerically. By analyzing of the coefficients mentioned rigorous results on orbital stability or instability are obtained.

In the case of oscillations with small amplitudes as well as in the case of rotations with high angular velocities the problem of orbital stability is studied analytically.