Reduction of Forward Difference Operators in Principal G-bundles

Retraction maps on Lie groups can be successfully used in mechanics and control theory to generate numerical integration schemes, for ordinary differential equations with a variational origin, recovering at the same time a discrete version of the energy and symplectic structure conservation properties, that are characteristic of smooth variational mechanics. The present work fixes the specific tool that plays in gauge field theories the same role as retraction maps on geometric mechanics. This tool, the covariant reduced projectable forward difference operator, can be used for a covariant discretization of the main elements of a variational theory: the jet bundle, the Lagrangian density and the associated action functional. Particular interest is dedicated to the trivialized formulation of a gauge field theory, and its reduction into a theory where fields are given as principal connections and $H$-structures. Main characteristics of the presented method are its covariance by gauge transformations and the commutation of the discretization and the reduction processes