Abstract
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.
Highlights
Classical numerical algorithms formulated on a linear space can, in many cases, be generalized to an arbitrary manifold, if we fix some operator that allows to measure the difference between two elements of the manifold as an object on a linear space
We show that for the reduced theory, a reduced forward difference operator leads to a discretization process, identifying the connection bundle of the theory with a discrete version, and the H-reduced jet bundle with another discrete version, with the desired behaviour in the limit case for multi-points coalescing to a diagonal point
As the forward Jacobi (FJ) operator establishes a relation between the jet bundle JXY (ε) ∈ (JY) and the bundle JY ⊂ Y ×n, on the corresponding regular domains, we arrive to a discretization notion for Lagrangian densities, extending to the general case the notion of discretization of lagrangian densities that was introduced for mechanics in definition 3.1: Definition 5.12 For a fixed Lagrangian density LvolX described by a smooth volume form volX ∈ Ωn(X) and a smooth function L : JY → R, we call associated discrete Lagrangian determined by a projectable faithful forward difference (FD) operator ∆ on Y the following function: Ld = (L ◦ JXY ) · : jx0 y ∈ (J Y) → R
Summary
Classical numerical algorithms formulated on a linear space can, in many cases, be generalized to an arbitrary manifold, if we fix some operator that allows to measure the difference between two elements of the manifold as an object on a linear space. This article takes the most general gauge field theory (not necessarily on a trivial bundle) and explores a general covariant mechanism to discretise a variational principle and its possible reduction by a symmetry group, extending to principal bundles ideas that have been originally formulated for the reduction to discrete Euler-Poincaré equations in mechanics on a Lie group. Besides presenting and relating all these areas, are described : We give a trivialization, in terms of principal connections, of the jet bundle and of its H-reduction (proposition 4.5 and corollary 4.1), introducing and describing the properties of forward difference operators on fibered manifolds, and of its associated objects, the n-Jacobian map, which allows to discretize volume forms, and the Forward Jacobi operator, which identifies the jet bundle with a multi-point manifold (Theorem 5.1) allowing to discretize lagrangian densities (definition 5.12). The present work shows that a single object, the faithful, projectable, reduced forward difference operator, represents a central element in the discretization of variational gauge field theories, leading to mechanisms that relate reduced or un-reduced, smooth or discrete gauge fields, in a compatible way, and preserving the available symmetries
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
