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

TL;DR

This paper develops a reduction method for symplectic principal b1-bundles with momentum maps, enabling the analysis of non-autonomous Hamiltonian systems and describing Poisson structures on quotient spaces.

Contribution

It introduces a novel reduction procedure for symplectic principal b1-bundles, extending geometric Hamiltonian theory to non-autonomous systems.

Findings

Reduction yields Poisson structures on quotient manifolds.

Symplectic leaves correspond to coadjoint orbits.

Applicable to non-autonomous Hamiltonian systems.

Abstract

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.