Plan (many other possibilities!):
There is an elegant solution to the equations for $\vb{L}$ involving the Jacobi elliptic functions - doubly periodic analytic functions that generalize trig and exponential functions.
A rigid pendulum with an amplitude near 180 degrees turns out to be an exact analog of the spinning T-handle.
Finally, one of my favorite map projections - the Peirce quincuncial - is based on elliptic functions.