Special relativity allows for many orbital trajectories that are unfamiliar to classical mechanics . In this project, these orbits will be simulated for a two body system with a potential that is proportional to 1/r. We were able to produce simulations for the one-body case that matched the analytical solutions obtained by Dr. Boyer and conserved energy to an acceptable level. Additionally the chosen integration method was able to produce accurate results for some two-body orbits with a large mass ratio. However, when this integration technique was applied to the general two-body case it failed to reproduce classical trajectories for the non-relativistic limit and did not conserve energy to an acceptable level for most relativistic orbits.