Linearization: Laplace vs. Stiefel |
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| Authors: | André Deprit Antonio Elipe Sebastián Ferrer |
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| Institution: | (1) Grupo de Mecánica Espacial, Universidad de Zaragoza, 50009 Zaragoza, Spain;(2) Present address: National Institute of Standards and Technology, 20899 Gaithersburg, MD, U. S. A. |
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| Abstract: | The method for processing perturbed Keplerian systems known today as the linearization was already known in the XVIIIth century; Laplace seems 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 formalism of Matrix Theory to proceed exclusively in the context of quaternion algebra; on the other hand, we explain how, in the hierarchy of hypercomplex systems, both the KS-transformation and the classical projective decomposition emanate by doubling from the Levi-Civita transformation. We propose three ways of stretching out the projective factoring into four-dimensional coordinate transformations, and offer for each of them a canonical extension into the moment space. One of them is due to Ferrándiz; we prove it to be none other than the extension of Burdet's focal transformation by Liouville's technique. In the course of constructing the other two, we examine the complementarity between two 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 of lowering dimensions through a Reduction induced by ignoration of variables. |
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| Keywords: | Linearization KS-transformation Keplerian problem Canonical transformations Hamiltonian systems |
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