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

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.

35 pages

Cited by in corpus (5)

• A survey on cosymplectic geometry
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• Integrable systems in cosymplectic geometry
• On Locally Conformally Cosymplectic Hamiltonian Dynamics and Hamilton-Jacobi Theory
• Cosymplectic geometry, reductions, and energy-momentum methods with applications