Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance properties of the associated Euler-Poincar'e equations with advected parameters.