The development of the Poincaré-similar elements with eccentric anomaly as the independent variable

A new set of element differential equations for the perturbed two-body motion is derived. The elements are canonical and are similar to the classical canonical Poincaré elements, which have time as the independent variable. The phase space is extended by introducing the total energy and time as canonically conjugated variables. The new independent variable is, to within an additive constant, the eccentric anomaly. These elements are compared to the Kustaanheimo-Stiefel (KS) element differential equations, which also have the eccentric anomaly as the independent variable. For several numerical examples, the accuracy and stability of the new set are equal to those of the KS solution. This comparable accuracy result can probably be attributed to the fact that both sets have the same time element and very similar energy elements. The new set has only 8 elements, compared to 10 elements for the KS set. Both sets are free from singularities due to vanishing eccentricity and inclination.