The simplest analytical model to be used in the propagation of an orbit is the theory of the Keplerian motion of a celestial body. This theory, which is essential from several points of view, becomes hardly useful when perturbations are involved in the dynamics in the sense that analytical solutions are no longer available. There is a large number of perturbations which can be acting on satellites. At first sight and due to the large number of theories and procedures developed in celestial mechanics there would seem to be numerous analytical formulations for solving perturbed motion. Would it be possible to obtain an analytical solution for each particular case? The answer depends on the forces we model; due to the complex nature of the equations representing the physical models exactly integrable expressions are difficult to obtain. The search of analytical solutions usually relies on series expansions to express the motion (a source of practical difficulty). In some cases, the effects over time of given perturbations can be classified through the secular, short-periodic, and long-periodic variations of the classical orbital elements. Although these distinctions help us decide which effects to model, the practical difficulty of an infinite series still remains.
A classical approach to the many-body problem is that of using special perturbations. Nowadays and due to the availability of high-speed computers is an essential tool in Space Dynamics which exhibits a great advantage: it is applicable to any orbit involving any number of bodies and all sorts of astrodynamical problems, especially when these problems fall into regions in which general perturbation theories are absent. One such case is, for example, that NEO’s dynamics. The main disadvantage of this method is that it rarely leads to any general formulae; in addition, the body’s positions at all intermediate steps must be computed in order to arrive at the final configuration. Some classical special perturbation methods are: 1) the Cowell’s method and 2) the Encke’s method.