Reduction of forward difference operators in principal G-bundles

Ana Casimiro, César Rodrigo

Research output: Contribution to journalArticle

3 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)42-85
Number of pages44
JournalStatistics, Optimization and Information Computing
Volume6
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Fingerprint

Difference Operator
Gages
Gauge Field Theories
Mechanics
Bundle
Retraction
Discretization
Jet Bundle
Lie groups
Gauge Transformation
Symplectic Structure
Electric commutation
Control theory
Control Theory
Ordinary differential equations
Numerical integration
Conservation
Ordinary differential equation
Formulation
Energy

Keywords

  • Discretization
  • Euler-Poincaré Equations
  • Forward Difference Operators
  • Reduction
  • Variational Principles

Cite this

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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.",
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Reduction of forward difference operators in principal G-bundles. / Casimiro, Ana; Rodrigo, César.

In: Statistics, Optimization and Information Computing, Vol. 6, No. 1, 01.01.2018, p. 42-85.

Research output: Contribution to journalArticle

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