The Kapitza pendulum is a model in classical mechanics that exhibits counterintuitive behaviour. It is a pendulum with a pivot point that oscillates vertically.
If $\phi$ is the angle the pendulum have with respect to the vertical downward position, for some values of the driving frequency and amplitude, it can be stable in the inverted position $\phi = \pi$, and unstable in the upright position $\phi = 0$.
This is the opposite of what happens for a normal pendulum, which has always a stable equilibrium point with $\phi=0$ and unstable one with $\phi=\pi$.
The equation of motion in the moving frame contains a fictitious force term $$ mL\ddot{\phi} = mg\sin\phi - Am\omega^2 \cos(\omega t) \sin \phi $$ that in our case is equivalent to $$ \ddot{\phi} = -\frac{1}{L}(g+A\omega^2\cos(\omega t))\sin \phi $$ Let us consider the dynamic variable as composed of a fast and a slow part $$ \phi = \phi_0 + \tilde{\phi} $$ Linearizing, the equation of motion of $\tilde{\phi}$ is $$ \ddot{\tilde{\phi}} = -\frac{1}{L}(g+A\omega^2\cos(\omega t))\sin\phi_0 \tilde{\phi} $$ that is a Mathieu equation. [to be continued…]