This work studies the dynamic behavior of solar energetic particles (SEPs, defined as protons and electrons with energies of 1 MeV to 1 GeV) by simulating their motion in model electromagnetic fields. Because of the hazards they pose to orbiting spacecraft and manned spaceflight operations, these species must be included in modern modeling efforts, including forthcoming space weather models. In this thesis, we describe an original computer program called the Solar Energetic Particle Tracer (SEPTR). As part of an operational computer model, SEPTR calculates the upper rigidity cutoffs for particles within an evolving magnetosphere. We use a feature of SEPTR to calculate late a global ionospheric grid of rigidity cutoffs using the DGRF 1980 magnetic field model. Rather than map rigidity space in order to find the cutoffs, we use an algorithm designed to locate the upper cutoff in a minimum of time, which is more appropriate for an operational model. Comparisons to similar calculations by Smart and Shea [1985] support the validity of our method. A second feature of SEPTR is the ability to use data of SEP or X-ray fluxes in order to generate differential energy spectra for magnetospheric protons. The rigidity cutoffs are then applied to the proton spectrum in order to provide a representation of the local flux. Because we calculate actual particle trajectories rather than making use of popular approximations, our code is ideal for studying the limitations of adiabatic theory. We examine the expansion of the first adiabatic invariant series in terms of ordering parameter, identify examples of non-adiabatic behavior, and specify a form of the first adiabatic invariant that is appropriate for full particle tracing. The dayside magnetopause frequently has a geometry with a minimum in magnetic field strength off the magnetic equatorial plane. This feature leads to particle drift paths that move into the Northern or Southern polar cusp. Our model suggests a possible concentration of energetic electrons in these regions as well as chaotic radial diffusion when the particle resumes equatorial mirroring.