Swinging in Imaginary Time More on the Not-So-Simple Pendulum

When the small angle approximation is not made, the exact solution of the simple pendulum is a Jacobian elliptic function with one real and one imaginary period. Far from being a physically meaningless mathematical curiosity, the second period represents the imaginary time the pendulum takes to swing upwards and tunnel through the potential barrier in the semi-classical WKB approximation(1) in quantum mechanics. The tunneling here provides a simple illustration of similar phenomena in Yang{Mills theories describing weak and strong interactions.