From physics we know that v_y = d theta/dt sin(theta). From our initial ODE this is just g/L d2 theta/dt2 So minimization occurs at g/L d3 theta / dt3 !!

Outstanding! I totally missed this! My only issue is that if it works out, it might make the problem simpler. 😝

Great choice of BCs! The problem with an initial velocity is that you have to wait longer and longer for the pendulum to fall. I didn't have a great intuition for these Elliptical functions, so I just settled for a numerical solution at some point lol. This notebook is definitely clarifying though!

Ah yeah, I think you could have instead worked out the right amount of energy to start at the bottom with just enough speed to reach the top, then once you pass the max horizontal velocity, who cares how long it takes to reach the top?