Variational integrators for reduced field equations
AbstractIn 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-Poincaré 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-Poincaré equations. We present a numerical integration algorithm for discrete Euler-Poincaré equations, that extends integration algorithms for Euler-Poincaré 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.
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