Reduction of Forward Difference Operators in Principal G-bundles

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.

Discrete formulation for the dynamics of rods deforming in space

“…In the present work we shall follow the specific case of the dynamical equations of hyperelastic rods as leitmotiv to report certain results concerning the discretization of variational principles in principal bundles. These results were described by the authors in [10] and [11]. * Supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações) † Supported by Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/04561/2013 (Centro de Matemática, Aplicações Fundamentais e Investigação Operacional of Universidade de Lisboa CMAF-CIO).…”

“…On the other hand the specific expression of the discrete Lagrangian in terms of (q x , ψ x ) ∈ Γ(HStr x ) × Γ(End 1 P x ) or in terms of p x ∈ Γ(P x ) is complicated and strongly depends on the particular choice of ∆ At . Moreover, for these fields the specific form of discrete Euler-Lagrange equations (10) or Euler-Poincaré equations (11) imply the addition of several partial derivatives of the discrete Lagrangian, which makes the local coordinate expressions almost impossible to explore.…”

“…It relies on a generalization for principal bundles of the notion of retraction mapping for Lie groups. This generalization is the notion of reduced forward difference operator ∆ At given in definition IV.1, which fixes, for any facet β ∈ V n , a one-to-one correspondence (see [10]) between elements in the reduced jet space RJP xβ , and elements (q x0 , χ x0 ) ∈ HStr x0 × CP x0 for x 0 = π 0 (xβ). The second main ingredient is a very simple cubature rule for a simplicial domain, namely the action functional [xβ] ℓ(x, q(x), χ(x))vol X on a simplicial domain [xβ] ⊂ X with vertices xβ ∈ X n is approximated taking the value of the lagrangian ℓ(x 0 , q(x 0 ), χ(x 0 )) at a given point x 0 = π 0 (xβ) = xv 0 ∈ X, multiplied with the volume of the simplicial domain [xβ] vol X .…”

Variational integrators for reduced field equations

“…In a recent work by the authors [6], we have described a procedure to discretize the key element of a variational principle: its Lagrangian density. This work shows how the discretization mechanism can be performed in such a way that it commutes with a possible reduction, by the action of some group of symmetries H. This leads to get a covariant way to define discrete H-reduced Lagrangian densities, it remains to explore how such an object generates a variational principle that represents the discrete analogue of Euler-Poincaré reduction, developed in the smooth case in [6].…”

“…The work begins in section 2 with the presentation of the main tools of Euler-Poincaré reduction, including an introduction to the variational principles that lead to smooth Euler-Poincaré equations, with the relevant Theorem 2.2 that relates the original and reduced variational principles. This section also includes fundamental results obtained in [6], related to our problem. Next section 3 presents our discretization choice for the space, an abstract simplicial complex arising from the Coxeter-Freudenthal-Kuhn simplicial partition of a cartesian lattice.…”