In this paper, a new and more general set of slow equations of motion for computing gyrotron (including high-harmonic gyrotron) and peniotron (including both peniotrons, where the orbit is centered on the axis and gyropeniotrons) interactions is derived. These slow equations are convenient to use and allow the interactions to be investigated numerically. They reduce to the correct expressions in the small-signal limit to yield starting currents. Small-signal and large-signal calculations yield results that agree with those of other workers. The equations are used to investigate the gyropeniotron.