  Quantum optomechanics offers the potential to investigate quantum effects in
macroscopic quantum systems in extremely well controlled experiments. In this
paper we discuss one such situation, the dynamic stabilization of a mechanical
system such as an inverted pendulum. The specific example that we study is a
"membrane in the middle" mechanical oscillator coupled to a cavity field via a
quadratic optomechanical interaction, with cavity damping the dominant source
of dissipation. We show that the mechanical oscillator can be dynamically
stabilized by a temporal modulation of the radiation pressure force. We
investigate the system both in the classical and quantum regimes highlighting
similarities and differences.
