Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products

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.