Bundles and geometric structures associated with gyroscopic systems

The overview is devoted to topological and geometric structures associated with gyroscopic systems whose action functional S is multivalued. The expediency of constructing and studying them 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 {ie828-01} 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 {ie828-02} 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. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.