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 CMFD, 2007, Volume 22, Pages 100–126 (Mi cmfd86)

Bundles and Geometric Structures Associated With Gyroscopic Systems

E. I. Yakovlev

Abstract: The overview is devoted to topological and geometric structures associated with gyroscopic systems whose action functional $S$ is multivalued. The expediency of their constructing and studying is in particular stipulated by the fact that the standard methods of the calculus of variations in the problem with fixed endpoints are not effective for such functionals. One of the methods for overcoming the difficulties arising here is the application of bundles, foliations, connections, and also Riemannian and Lorentz manifolds. In this way, it turns out to be possible to perform the reduction of the two-point problem for $S$ to problems with fixed initial point and movable endpoint for the length functional ${\mathcal L}^*$ of a pseudo-Riemannian manifold foliated over the configurational space of the gyroscopic system considered. As the endpoint manifolds, the leaves of the Riemannian foliation are used, and the correspondence between the extremals of the functionals $S$ and ${\mathcal L}^*$ is stated by using the Ehresmann connection of this bundle. The paper discusses the results on the motions of natural mechanical systems with gyroscopic forces and gyroscopic systems of relativistic type obtained by using the above reduction and also the topological and geometric constructions used in it.

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English version:
Journal of Mathematical Sciences, 2008, 153:6, 828–855

Bibliographic databases:

UDC: 514.83

Citation: E. I. Yakovlev, “Bundles and Geometric Structures Associated With Gyroscopic Systems”, Geometry, CMFD, 22, PFUR, M., 2007, 100–126; Journal of Mathematical Sciences, 153:6 (2008), 828–855

Citation in format AMSBIB
\Bibitem{Yak07} \by E.~I.~Yakovlev \paper Bundles and Geometric Structures Associated With Gyroscopic Systems \inbook Geometry \serial CMFD \yr 2007 \vol 22 \pages 100--126 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd86} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2336509} \zmath{https://zbmath.org/?q=an:1157.58304} \elib{https://elibrary.ru/item.asp?id=13593490} \transl \jour Journal of Mathematical Sciences \yr 2008 \vol 153 \issue 6 \pages 828--855 \crossref{https://doi.org/10.1007/s10958-008-9147-7} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-54249142080} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. Y. Zinchenko, E. I. Yakovlev, “Smooth almost $\Delta$-fiber bundles over simplicial complexes”, Russian Math. (Iz. VUZ), 54:11 (2010), 1–17
2. E. I. Yakovlev, “Two-point boundary value problem for gyroscopic systems in some Lorentzian manifolds”, Russian Math. (Iz. VUZ), 57:6 (2013), 53–61
3. E. I. Yakovlev, T. A. Gonchar, “Geometry and topology of some fibered Riemannian manifolds”, Russian Math. (Iz. VUZ), 62:2 (2018), 69–85
4. E. I. Yakovlev, T. A. Gonchar, “Causal Properties of Fibered Space-Time”, Math. Notes, 106:1 (2019), 118–132
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