# American Institute of Mathematical Sciences

December  2019, 11(4): 539-552. doi: 10.3934/jgm.2019026

## The problem of Lagrange on principal bundles under a subgroup of symmetries

We dedicate this work to our friend Darryl D. Holm on the occasion of his 70th birthday.

Received  April 2018 Revised  May 2019 Published  November 2019

Fund Project: This work has been partially supported by MICINN (Spain) under projects MTM2015-63612-P and PGC2018-098321-B-I00, as well as Consejería de Educación, Junta de Castilla-León (Spain) under project SA090G19.

Given a Lagrangian density $L{\bf{v}}$ defined on the $1$-jet extension $J^1P$ of a principal $G$-bundle $\pi \colon P\to M$ invariant under the action of a closed subgroup $H\subset G$, its Euler-Poincaré reduction in $J^1P/H = C(P)\times_M P/H$ ($C(P)\to M$ being the bundle of connections of $P$ and $P/H\to M$ being the bundle of $H$-structures) induces a Lagrange problem defined in $J^1(C(P)\times_M P/H)$ by a reduced Lagrangian density $l{\bf{v}}$ together with the constraints ${\rm{Curv}}\sigma = 0, \nabla ^\sigma \bar{s} = 0$, for $\sigma$ and $\bar{s}$ sections of $C(P)$ and $P/H$ respectively. We prove that the critical section of this problem are solutions of the Euler-Poincaré equations of the reduced problem. We also study the Hamilton-Cartan formulation of this Lagrange problem, where we find some common points with Pontryagin's approach to optimal control problems for $\sigma$ as control variables and $\bar{s}$ as dynamical variables. Finally, the theory is illustrated with the case of affine principal fiber bundles and its application to the modelisation of the molecular strands on a Lorentzian plane.

Citation: Marco Castrillón López, Pedro Luis García Pérez. The problem of Lagrange on principal bundles under a subgroup of symmetries. Journal of Geometric Mechanics, 2019, 11 (4) : 539-552. doi: 10.3934/jgm.2019026
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