The Lagrange-Poincare equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin-Noether Theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, and molecular strands illustrate various aspects of the theory. (C) 2011 Elsevier B.V. All rights reserved.