Poisson-Poincar\'e reduction for Field Theories

TL;DR

This paper develops a Poisson-Poincaré reduction framework for Hamiltonian field theories with symmetry, linking it to Lagrange-Poincaré reduction and demonstrating its application to a charged strand model.

Contribution

It introduces a novel reduction method for covariant Hamiltonian field theories with symmetry, extending classical Poisson reduction to the field theory context.

Findings

Established a Poisson covariant formulation for Hamiltonian systems on fiber bundles.

Derived a reduction procedure analogous to Poisson-Poincaré reduction for field theories.

Applied the reduction framework to a charged strand model in an electric field.

Abstract

Given a Hamiltonian system on a fiber bundle, there is a Poisson covariant formulation of the Hamilton equations. When a Lie group G acts freely, properly, preserving the fibers of the bundle and the Hamiltonian density is G-invariant, we study the reduction of this formulation to obtain an analogue of Poisson-Poincar\'e reduction for field theories. This procedure is related to the Lagrange-Poincar\'e reduction for field theories via a Legendre transformation. Finally, an application to a model of a charged strand evolving in an electric field is given.