Classical Perturbation Theory in Celestial Mechanics. The Equations of Planetary Motion

Abstract

In this chapter, we mainly aim to give an idea of the infancy and the adolescence of perturbation theory. Keeping in mind the general assumption according to which the whole book is built up, we have tried, as far as possible, to account for the developments and the changes which perturbation theory has undergone in the long term. This is almost always the only way to understand a problem thoroughly: that is, to understand why, at a certain instant in history, the problem is stated in a particular way and what are the missing answers and the reason it is not possible to answer certain questions hic et nunc. As the reader can see, in perturbation theory, although with difficulty but ineluctably, the idea that one can, nay must, average over short-period terms has won. This enables one to obtain results valid over the long period, that is, results of “secular” validity. It has been remarked that here, as in other fields, the idea has just been accepted that the end justifies the means (instead of the moral justification, here the mathematical proof is missing); we shall come back to this subject at the end of the whole presentation of the theory. If the introduction of the averaging procedure removes the so-called secular terms, nonetheless it does not remove the other plague of perturbation theory: the appearance of small denominators and related problems of convergence of the perturbative series. We shall see about these in the next chapter. After having explained the classical non-canonical methods and briefly mentioned the more modern averaging method, we also give an example of a problem, previously dealt with following a perturbative approach, which has been recognized to be solvable analytically in closed form.