Geometric dynamics on the automorphism group of principal bundles:  Geodesic flows, dual pairs and  chromomorphism groups

François Gay-Balmaz; Cesare Tronci; Cornelia Vizman

doi:10.3934/jgm.2013.5.39

Article Contents

2013, Volume 5, Issue 1: 39-84. Doi: 10.3934/jgm.2013.5.39

This issue Previous Article Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems Next Article Vector fields with distributions and invariants of ODEs

Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups

1.
Laboratoire de Météorologie Dynamique, École Normale Supérieure/CNRS, F-75231 Paris
2.
Department of Mathematics, University of Surrey, Guildford GU2 7XH
3.
West University of Timişoara, RO-300223 Timişoara

Received: June 2012

Revised: January 2013

Published: March 2013

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Abstract

We formulate Euler-Poincaré equations on the Lie group $Aut(P)$ of automorphisms of a principal bundle $P$. The corresponding flows are referred to as EP$Aut$ flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle $P$, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing $\delta\text{-like}$ momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group $Aut_{vol}(P)$ of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.

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Mathematics Subject Classification: Primary: 37K65, 58D05; Secondary: 53D20.

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