Orbit generation by use of Taylor's expansion

Feynman (I) presents a method of tracing the orbit of a particle in a field of force. He makes each step of the orbit by calculating an average velocity for the particle based on its initial velocity and the acceleration it is subjected to over a finite length of time, At, and calculating the new location of the particle, assuming it moves in a straight line. Since the acceleration of the particle is known as a function of position, it is possible to use Taylor's expansion to get a higher order approximation. The question arises as to what is the most efficient way to plot an orbit over a cycle: should we use a few steps and a higher order of approximation, or many steps with the lowest order of approximation. It must be realized that the higher the order of approximation the more CPU time is required per step, but the steps may be larger to achieve the same degree of accuracy in the complete orbit; similarly, the more steps used, the more CPU is required. Several orders of approximations were used on Feynman's problem to decide which order leads to the least CPU time for equal accuracy in the determination of the orbit's period. It was found that~the third order approximation had almost twice the efficiency of any order from