Euler–Poincaré reduction in principal bundles by a subgroup of the structure group

Given a Lagrangian density Lv defined on the 1-jet bundle J1PJ1P of a principal GG-bundle π:P→Mπ:P→M invariant with respect to a subgroup HH of GG, the reduction of the variational problem defined by Lv to (J1P)/H=C×M(P/H)(J1P)/H=C×M(P/H), where CC is the bundle of connections in PP, is studied. It is shown that the reduced Lagrangian density lv defines a zero order variational problem on connections σσ and HH-structures s̄ of PP with non-holonomic constraints Curvσ=0 and ∇σs̄=0 and set of admissible variations those induced by the infinitesimal gauge transformations in CC and P/HP/H. The Euler–Poincaré equations for critical reduced sections are obtained as well as the reconstruction process to the unreduced problem. The corresponding conservation laws and their relationship with the Noether theory are also analyzed. Finally, some instances are studied: the heavy top and affine principal bundles, the main application of which is used for molecular strands.