Determining trajectories of objects in three-dimensional space is fundamental to many disciplines. Whether the objects be molecules or celestial bodies, the standart Newtonian equations of motion describe this dynamic behavior.. Unfortunately, however, the complexity of a real system normally requires that the equations of motion be solved numerically and methods presented in classical mechanics textbooks are of little practical use. In addition, the standart equations of motion involving Eulaer angles contain singularities and are therefore not suitable for computer solution. This paper discusses implementation of two powerful but little known techniques using quaternions, the Evans and the Rapaport methods. These little known techniques have been developed by and used for molecular modeling, but their application is far reaching, and can be applied to any system where quantum or relativistic effects may be neglected. These methods shoe excellent conservation of energy over many millions of time steps and are free of singularities. Herein we compare the two methods in terms of efficiency and accuracy. Due to their general nature, the algorithns may be readily coded to meet the needs of many applications.