It appears that a classical example of a "monodromy" has been demonstrated for the first time.
When a spherical pendulum is vertically upside down, even a tiny deviation will kick it out of position. This type of unstable point in a mechanical system corresponds to a singularity in the constants of motion.
Mathematicians call the study of a system as it loops around such a singularity “monodromy,” which literally means “once around.” While the effects of monodromy should appear both in classical systems, such as pendulums and spinning tops, and quantum systems, such as rotating molecules, they have only been observed experimentally (and rarely) in a quantum system. Writing in Physical Review Letters, Noah Fitch and colleagues at JILA and the University of Colorado, US, in collaboration with the University of Sydney, Australia, present the first experimental demonstration of monodromy in a classical system.
The link above also gives a free access to the PRL paper.
So what is the example of a classical monodromy? A spherical pendulum in which the arm is a spring.
This looks familiar to me because I could have sworn that I saw something like this in my qualifying exam! :)