Integrable Hamiltonian systems are said to display nontrivial monodromy if fundamental action-angle loops defined on phase-space tori change their topological structure when the system is carried around a circuit. In an earlier paper it was shown that this topological change can occur as a result of time evolution under certain rather abstract flows in phase space. In the present paper, we show that the same topological change can occur as a result of application of ordinary forces. We also show how this dynamical phenomenon could be observed experimentally in classical or in quantum systems.