The document analyzes the motion of an inverted pendulum with an oscillating support. It uses Lagrangian mechanics to derive an equation of motion for the pendulum's angle θ. For small oscillations of θ, the equation is approximated as θ̈ + θaω2cos(ωt) - ω02 = 0. This shows the pendulum's motion depends on the frequency and amplitude of the support oscillations. If the support frequency ω is high enough, the pendulum undergoes stable oscillations instead of falling over. An analytical approximation of θ as a small oscillation correlated with the support motion is derived, explaining this stabilizing effect.