Classical periodic trajectories are shown to exist for a test particle moving in an external SU(2) gauge field. The field is assumed to be static and of the form proposed by Wu and Yang. The energy of the system is positive, so the orbits are expected to give rise to a spectrum of metastable states. A Bohr-Sommerfeld quantization scheme is proposed for finding the spectrum. The results are applied to the problem of a particle interacting with a Polyakov-'t Hooft monopole, via the explicit solution of Prasad and Sommerfield. © 1977 The American Physical Society.