Reduction of symplectic principal $\mathbb{R}$-bundles

Details Reduction of symplectic principal $\mathbb{R}$-bundles Reduction of symplectic principal $\mathbb{R}$-bundles

2012-01-23

arxiv.org/abs/1201.4690v2

Mathematics

Differential Geometry

35 pages

Scientific paper

We describe a reduction process for symplectic principal $\mathbb{R}$-bundles in the presence of a momentum map. This type of structures plays an important role in the geometric formulation of non-autonomous Hamiltonian systems. We apply this procedure to the standard symplectic principal $\mathbb{R}$-bundle associated with a fibration $\pi:M\to\mathbb{R}$. When $\pi$ is a principal $G$-bundle and $G_\nu$ denotes the isotropy group associated with an element $\nu$ in the dual to the Lie algebra of $G$, we use the reduction process in order to describe a Poisson structure on the quotient manifold $M/G_\nu$ whose symplectic leaves are isomorphic to the coadjoint orbit $\mathcal{O}_\nu$ . Moreover, we show a reduction process for non-autonomous Hamiltonian systems on symplectic principal $\mathbb{R}$-bundles.

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