We have derived relativistically correct gyro- and bounce-averaged Hamiltonian equations of motion to describe the motions of non-isotropic proton distributions in the Earth's inner and middle magnetosphere. We have focused on the case of equatorially-mirroring protons, and we have modified the Magnetospheric Specification Model (MSM) to trace these particles. We call the new particle simulation the Radiation Belt Test Code (RBTC). We have modeled the March 1991 magnetic storm, which was an extremely strong magnetospheric compression in which protons were energized to 1 to 100 MeV on time scales of a few minutes. We have compared our results with CRESS data collected during the event, and with the simulation results of Hudson et al. (3). We see a significant flux increase, but it is not as large as the increase observed by CRRES. We conclude that our model correctly describes the gross features of high-energy magnetospheric protons, but that the present algorithm of the MSM is too computationally intensive to model these equations in a reasonable time, especially for the highest energy particles that we were interested in. Suggestions for improvements and alternative methods are suggested.