PACS 45.50.-j -Dynamics and kinematics of a particle and a system of particles PACS 45.05.+x -General theory of classical mechanics of discrete systems PACS 03.65.Vf -Phases: geometric; dynamic, topological Abstract -Hamiltonian monodromy -a topological property of the bundle of regular tori of a static Hamiltonian system which obstructs the existence of global action-angle variables-occurs in a number of integrable dynamical systems. Using as an example a simple integrable system of a particle in a circular box with quadratic potential barrier, we describe a time-dependent process which shows that monodromy in the static system leads to interesting dynamical effects.