Symmetric Multifrequency Oscillations of a Lagrangian System and Their Stabilization

This paper considers a Lagrangian system subjected to positional forces. By assumption, the frequencies of small oscillations for an equilibrium are incommensurable. The existence of multifrequency symmetric oscillations in the neighborhood of the equilibrium and their extension to global families of such oscillations is established; the families fill the entire domain of the phase space adjacent to the equilibrium. A nonlocal continuation of the above families in the system parameter is proved. In the conservative case, an autonomous control is found, and a symmetric multifrequency oscillation is stabilized in the large. In this case, feedback on potential energy is constructed.