Lie-Poisson reduction in principal bundles by a subgroup of the structure group

We study Hamiltonian field theories on the multisymplectic bundle of a principal G-bundle with Hamiltonian densities invariant under a subgroup $H\subset G$. Using the covariant bracket formulation, we reduce the polysymplectic space and derive the corresponding reduced observables, brackets, and equations of motion, yielding a Lie--Poisson reduction by a subgroup for field theories. We also address the reconstruction problem, characterizing reconstruction in terms of the flatness of an associated connection. Several examples, including the heavy top, molecular strands with broken symmetry, and affine principal bundles, illustrate the general framework.