Use of the Geometric Elements in Numerical Simulations

We derive the transformations to convert the state vector in cartesian coordinates into geometric orbital elements (and conversely the geometric elements into the state vector) for a test particle moving around an oblate planet. These transformations arise from the epicyclic theory and are accurate to second order in eccentricity and inclination. This paper is written to be directly used for computational purposes, such as the numerical study of ring dynamics.