Motion in the field of a rotationally symmetric potential: Exact use of an approximate equation for the derivative of the field of directions along the normal to a trajectory

The previously derived equation (Agekyan 1974) for the derivative ?f/?n of the field of directions along the normal to a trajectory is approximate, because differentiating along the normal takes the point out of the orbit and changes the third integral of motion. However, on the envelope of the trajectory, i.e., on the contour of an orbit or a fold, ?f/?n undergoes a discontinuity of the second kind. Many authors have used this property to find points of the contours of orbits and folds. Although the integrable equation is approximate, the envelope points are determined accurately.