There is a well developed and useful theory of Hamiltonian reduction for
semidirect products, which applies to examples such as the heavy top,
compressible fluids and MHD, which are governed by Lie-Poisson type equations.
In this paper we study the Lagrangian analogue of this process and link it with
the general theory of Lagrangian reduction; that is the reduction of
variational principles. These reduced variational principles are interesting in
their own right since they involve constraints on the allowed variations,
analogous to what one finds in the theory of nonholonomic systems with the
Lagrange d'Alembert principle. In addition, the abstract theorems about
circulation, what we call the Kelvin-Noether theorem, are given.