Monodromy literally means "once around." The term is applied in mathematics to systems that run around a singularity. In these systems, a parameter that describes the state of the system changes when the system loops around the singularity. Specifically, Hamiltonian monodromy is a topological obstruction to the existence of global action angle variables. Since monodromy’s discovery in 1980, mathematicians have predicted that many physical systems have it, including pendulums and tops as well as atoms and molecules.  We demonstrated the first experimental confirmation of classical monodromy.   

The elastic pendulum, which consists of a mass on a spring in a gravitational field, was predicted to to have monodromy. To begin the motion, the mass is pulled down and let go such that it starts with nearly-pure bounching motion. After some time the motion transforms into nearly-pure pendular swinging. This transfer of energy happens very efficeantly becauce the frequencies of bounching to swinging are in a ratio of 2:1. The manifestation of monodromy in this system is the non-periodic behavior of the rotation number. In this example, the stepwise precession anlge of the swing ellipse is a rotation number.