LINEARIZATION - LAPLACE VS STIEFEL

The method for processing perturbed Keplerian systems known today as t he linearization was already known in the XVIII(th) century; Laplace s eems to be the first to have codified it. We reorganize the classical material around the Theorem of the Moving Frame. Concerning Stiefel's own contribution to the question, on the one hand, we abandon the form alism of Matrix Theory to proceed exclusively in the context of quater nion algebra; on the other hand, we explain how, in the hierarchy of h ypercomplex systems, both the KS-transformation and the classical proj ective decomposition emanate by doubling from the Levi-Civita transfor mation. We propose three ways of stretching out the projective factori ng into four-dimensional coordinate transformations, and offer for eac h of them a canonical extension into the moment space. One of them is due to Ferrandiz; we prove it to be none other than the extension of B urdet's focal transformation by Liouville's technique. In the course o f constructing the other two, we examine the complementarity between t wo classical methods for transforming Hamiltonian systems, on the one hand, Stiefel's method for raising the dimensions of a system by means of weakly canonical extensions, on the other, Liouville's technique o f lowering dimensions through a Reduction induced by ignoration of var iables.