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Contact geometry
It is not possible to endow an odd-dimensional with the symplectic structure. The analogue of the symplectic structure in this case is little less symmetric, however outstanding structure called contact structure. The source of this in mechanics are phase spaces (or cotangent bundles to configurational manifolds) on which there always exists the canonical symplectic structure. The source of contact structures are manifolds of contact elements of configuration spaces. A contact elemet to an n-dimensional smooth manifold at a given point is a (n-1)-dimensional tangent plane to the manifold at this point (or a (n-1)-dimensional linear subspace of an n-dimensional tangent space at this pont). The set of all contact elements of an n-dimensional manifold has a natural smooth manifold structure of dimension (2n-1). It turns out that on this manifold with odd dimension there is an outstanding additional structure called contact structure. The manifold of contact elements of an n-dimensional Riemannian manifold is strictly connected to the (2n-1)-dimensional manifold of unit tangent vectors to this manifold, or with (2n-1)-dimenional isoenergetic manifold of a material point moving inertially on the Riemannian manifold. On the other hand, contact structures on these (2n-1)-dimensional manifolds are connected to the symplectic structure in (2n)-dimensional phase space of this point (or in the cotanget bundle of the n-dimenional Riemannian manifold).
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