Noether's theorem in symplectic bundles (Q5941835)

The authors first discuss different kinds of symplectic bundles and their potential use in general relativity. They argue that the concept of symplectic fibration (which generalizes the notion of symplectic vector space) is somewhat too general for their purposes, and propose instead to work on seeded fibre bundles (SFB), as introduced by \textit{V. Liern} and \textit{J. Olivert} [C. R. Acad. Sci., Paris, Sér. I 320, 203-206 (1995; Zbl 0856.58013)]. Essentially, starting from a principal \(G\)-bundle \(\lambda=(P,B,\pi,G)\) with a connection, and a symplectic manifold on which \(G\) acts by symplectomorphisms, an associated fibre bundle \(\lambda(F)\) with fibre \(F\) is called a seeded fibre bundle, if there exists a projectable foliation \(S\) in the horizontal distribution, such that every fibre cuts each leaf of \(S\) in at most one point. A theorem is recalled which characterizes a SFB as a bundle of fibrations in which every fibre is the base manifold of another fibration whose total space is a presymplectic manifold. In applications, the base \(B\) is a space-time manifold (but \(G\) need not be the Lorentz or Poincaré group) and \(F\) somehow represents a dynamical system. The foliation which is needed for constructing a related SFB then must be provided with a `motion law'. The final section is about extending the concept of a dynamical group in order to come to a generalization of Noether's theorem in a SFB-structure. The idea thereby is that \(G\) should act as a dynamical group also in the presymplectic manifolds which cover the fibres.