The rotations of a pendulum excited by a high-frequency harmonic variation of its length

The motion of a mathematical pendulum, whose length is varied harmonically with time at a frequency that is high compared with the characteristic frequency of small oscillations of a pendulum of constant length, is considered. The method of canonical transformations and results obtained previously on Poincaré periodic motions in close to integrable Hamiltonian systems with one degree of freedom,1 are the basis of the investigation. It is shown that periodic motions of the pendulum exist that are close to its rotation with an angular frequency, the mean value of which over time is a multiple of the frequency with which the pendulum length changes. An explicit expression for the periodic motions is obtained in the first approximation with respect to the small parameter. The non-linear problem of the stability of Lyapunov periodic motions is solved.