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Work out the Lagrangian equations of motion for an inverted pendulum on a
moving base.

m1 mass of block

m2 mass of pendulum

l length of pendulum

x[t] position of m1

φ[t] angle of pendulum 0 is down

g force of gravity

Fx[t] the force we apply to the m1.

Fφ[t] the tangential force we apply to m2.

By inspection we find the  Lagrangian

We use x2 and y2 as the x and y position of m2 - now we need x2 and y2 in
terms of x, l and φ.

Now to get the Lagrangian equations of motion.