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

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.