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In this work, we develop a Lagrangian reduction theory for covariant fieldtheories with gauge symmetries. These symmetries are modeled by a Lie groupfiber bundle acting fiberwisely on a configuration bundle. In order to reducethe variational principle, we utilize generalized principal connections, a typeof Ehresmann connections that are equivariant by the fiberwise action. Afterobtaining the reduced equations, we give the reconstruction condition and werelate the vertical reduced equation with the Noether theorem. Lastly, weillustrate the theory with several examples, including the classical case(Lagrange-Poincar\'e reduction), Electromagnetism, symmetry-breaking andnon-Abelian gauge theories.

Gauge reduction in covariant field theory