Abstract
In the reduction of field theories in principal $G$-bundles, when a subgroup $H\subset G$ acts by symmetries of the Lagrangian, each of the $H$-reduced unknown fields decomposes as a flat principal connection and a parallel H-structure. A suitable variational principle with differential constraints on such fields leads to necessary criticality conditions known as Euler-Poincare equations. We model constrained discrete variational theories on a simplicial complex and generate from the smooth theory, in a covariant way, a discrete variational formulation of $H$-reduced field theories. Critical fields in this formulation are characterized by a corresponding discrete version of Euler-Poincare equations. We present a numerical integration algorithm for discrete Euler-Poincare equations, that extends integration algorithms for Euler-Poincare equations in mechanics to the case of field theories and, also, extends integration algorithms for Euler-Lagrange equations in discrete field theories to the case of variational principles with constraints. For regular reduced discrete Lagrangians, this algorithm allows to univocally propagate initial condition data, on an initial condition band, into a solution of the corresponding equations for the discrete variational principle.
Highlights
In the community working on numerical methods for ordinary and partial differential equations arising in physics, it has become clear in the last years that discretising the variational principles, that originally generated the equations, is a very fruitful method, when we are looking for some discrete version of the equations and integrators for this discrete version
In many other situations the corresponding field theory has a natural time variable and its discretization allows for a treatment similar to that of discrete mechanical systems [14, 15]
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
Summary
In the community working on numerical methods for ordinary and partial differential equations arising in physics, it has become clear in the last years that discretising the variational principles, that originally generated the equations, is a very fruitful method, when we are looking for some discrete version of the equations and integrators for this discrete version. 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]. As happened in non-reduced discrete field theories [5], the existence of an inverse of the momentum mapping (what we call an integrator) is the central element in an iterative algorithm, that allows to extend admissible H-reduced fields on some initial band to globally defined fields, that are critical for the discrete Euler-Poincaré variational principle. Certain technical aspects related to quotient manifolds and polytopes are included in the Appendix section
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