SOME PROBLEMS OF THE MOTION OF A PENDULUM WHEN THERE ARE HORIZONTAL VIBRATIONS OF THE POINT OF SUSPENSION

The motion of a mathematical pendulum whose point of suspension perfor ms small-amplitude horizontal harmonic oscillations is considered. The non-integrability of the equation of motion of the pendulum is establ ished. The periodic motion of the pendulum originating from a stable p osition of equilibrium is obtained and its stability is investigated. Unstable periodic motions originating from unstable positions of equil ibrium are indicated and the separatrice surfaces asymptotic to these motions are determined. The problem of the existence and stability of periodic motions of the pendulum originating from its oscillations wit h arbitrary amplitude and rotations with arbitrary mean angular veloci ty is investigated.