As you clearly realized, the best way to do this is to ignore Penrose's "hint", and take

as your coordinate right from the beginning.

The Hamiltonian method is slighly more complex in this case only because you first need to find out what

is.

The direct Newtonian calculation is much easier than you make it out to be - it's in fact the easiest method: Given that the pendulum length is fixed, you can ignore radial/centripetal forces altogether, and just calculate tangential acceleration due to (the tangiential component of) gravitational force:

is the

immediate result!