Why does Kapitza's pendulum oscillate upside down?

Using Lagrangian mechanics to understand how a pendulum driven vertically at high frequency and small amplitude can remain in stable equilibrium in an inverted configuration, and deriving the condition for stability.

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About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.

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