Reduction in principal fiber bundles: covariant Euler-Poincaré equations.

Abstract

In Lagrangian mechanics the simplest example of reduction is the Euler-Poincaré reduction. In this case, the configuration space is a Lie group G and the Lagrangian function L:TG→R is invariant under the natural action of the group on TG by right translations. Then L induces a function l:(TG)/G≅g→R, g being the Lie algebra of G, and the Euler-Lagrange equations defined by L transform into a new group of equations for l called the Euler-Poincaré equations. For example, this is the case for the dynamics of the rigid body. In the present paper, the authors extend the idea of reduction to Lagrangian field theory. In this framework, the analogous configuration is a principal bundle π:P→M with structure group G (a matrix group) and a Lagrangian L:J1P→R invariant under the natural action of G on the 1-jet bundle defined by (j1xs)⋅g=j1x(Rg∘s), where Rg denotes the right translation by g on P. Let l:(J1P)/G→R be the induced mapping. It is proved that the Euler-Lagrange equations define a group of equations for critical sections, which generalize the Euler-Poincaré equations of mechanics. As is well known, the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. This fact gives a geometrical meaning to the previous reduction. In particular, the critical sections of the Euler-Poincaré equations are sections of this bundle, and therefore they can be understood as principal connections of π:P→M. The authors exploit this idea in order to explain the compatibility conditions needed for reconstruction. The Euler-Poincaré equations do not suffice to reconstruct the Euler-Lagrange equations. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. This fact is characteristic of field theory and does not appear in classical mechanics. Finally, the authors study the Euler-Poincaré equations in two examples from the variational approach to harmonic mappings.