Classical Higgs fields

We consider a classical gauge theory on a principal fiber bundle $P\to X$ in the case where its structure group $G$ is reduced to a subgroup $H$ in the presence of classical Higgs fields described by global sections of the quotient fiber bundle $P/H\to X$. We show that matter fields with the exact symmetry group $H$ in such a theory are described by sections of the composition fiber bundle $Y\to P/H\to X$, where $Y\to P/H$ is the fiber bundle with the structure group $H$, and the Lagrangian of these sections is factored by virtue of the vertical covariant differential determined by a connection on the fiber bundle $Y\to P/H$.