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

Abstract Given a Lagrangian density L v defined on the 1-jet bundle J 1 P of a principal G -bundle π : P → M invariant with respect to a subgroup H of G , the reduction of the variational problem defined by L v to ( J 1 P ) / H = C × M ( P / H ) , where C is the bundle of connections in P , is studied. It is shown that the reduced Lagrangian density l v defines a zero order variational problem on connections σ and H -structures s of P with non-holonomic constraints Curv σ = 0 and ∇ σ s = 0 and set of admissible variations those induced by the infinitesimal gauge transformations in C and P / H . The Euler–Poincare 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.