Averaging method and asymptotic solutions in some mechanical problems

Abstract: In the paper we consider systems in oscillating force fields such that the classical method of averaging can be applied. We present sufficient conditions for the existence of forced oscillations in such systems and study the asymptotic behaviour of some solutions. In particular, we show that for an inverted pendulum with a horizontally moving pivot point in an oscillating gravity field there exists a solution along which the pendulum never falls.