In this paper we consider the system of Hamiltonian differential equations, which determines small oscillations of a dynamical system with n parameters. We demonstrate that this system determines an affinor structure J on the phase space TRⁿ. If J² = ωI, where ω = ±1,0, the phase space can be considered as the biplanar space of elliptic, hyperbolic or parabolic type. In the Euclidean case (Rⁿ = Eⁿ) we obtain the Hopf bundle and its analogs. The bases of these bundles are, respectively, the projective (n-1)-dimensional spaces over algebras of complex, double and dual numbers.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc57-0-15,
author = {Boris N. Shapukov},
title = {Affinor structures in the oscillation theory},
journal = {Banach Center Publications},
volume = {58},
year = {2002},
pages = {211-217},
zbl = {1038.53029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc57-0-15}
}
Boris N. Shapukov. Affinor structures in the oscillation theory. Banach Center Publications, Tome 58 (2002) pp. 211-217. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc57-0-15/