Kapitza Inverted Pendulum (2D)

A rod balances upside-down on a pivot shaken vertically. Same equation and numbers as the 3D lab.

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The science

With the pivot height y(t) = A cos(Omega t) the angle from straight up obeys theta'' = [(g - A Omega^2 cos(Omega t)) / L] sin(theta) - c theta'. Averaging over the fast drive gives an effective potential with a minimum at theta = 0 when (A Omega)^2 > 2 g L (small A/L). The slow motion about the inverted position then has omega_s^2 = A^2 Omega^2 / (2 L^2) - g / L.

Integrator: fixed-step RK4, 0.5 ms.
Simplifications: point bob on a massless rod, viscous damping on the angle only, no pivot friction. The criterion is a small-amplitude asymptotic result, so near the threshold the true boundary shifts slightly with A/L and damping.
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