  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.
