Maintenance of polarization of polarized protons in a linear accelerator is known to be feasible. Circular accelerators present a different problem, and the investigation of the interaction of orbit dynamics and particle polarization in general is undertaken. The equation of motion of the spin vector of a charged particle in a magnetic field as formulated by Bargmann, Michel, and Telegdi is utilized in the study of depolarization for several accelerators. High values of depolarization are obtained, and means for avoiding such depolarization are suggested. (D.C.W.)