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Lagrange-Poincaré reduction in affine principal bundles

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  • Given an $H$-principal bundle $Q\to M$ and a (left) linear action of $H$ to a real vector space $V$, let $E\to M$ be the vector bundle associated to $Q$ and to the linear action, and $Q\times_M E$ the affine principal bundle with structure group the semidirect group $G = H Ⓢ V$. If $L v$ is a Lagrangian density defined on the 1-jet bundle $J^1(Q\times_M E)$ invariant by the subgroup $H \hookrightarrow H Ⓢ V$, the variational problem induced on $(J^1(Q\times_ME)) /H = C(Q)\times_M J^1E$, where $C(Q)$ is the bundle of connections in $Q$, is considered. We show that the reduced Lagrangian density $lv$ defines a variational problem on connections $\sigma \in \Gamma (C(Q))$ and on sections $e\in \Gamma(E)$, with constraint $\textrm{Curv }\sigma =0$, and set of admissible variations those induced on $\Gamma (C(Q))$ by the infinitesimal gauge transformations of $Q$ and on $\Gamma(E)$ by arbitrary vertical variations. The Lagrange-Poincaré equations for the critical reduced sections are obtained, as well as the reconstruction process to the unreduced problem. The Poincaré equation is interpreted as the reduction of the Noether conservation law corresponding to the $H$-symmetry of the Lagrangian density $L v$. We also study the reduced system as a Lagrange problem through a suitable choice of the Lagrange multipliers. This allows us to establish a Hamilton-Cartan formalism for this class of systems. Finally, we discuss the molecular strands, a motivating example of the theory.
    Mathematics Subject Classification: Primary: 58E30; Secondary: 70S05, 53C05.


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