Higher order Lagrange-Poincar\'e and Hamilton-Poincar\'e reductions

Motivated by the problem of longitudinal data assimilation, e.g., in the registration of a sequence of images, we develop the higher-order framework for Lagrangian and Hamiltonian reduction by symmetry in geometric mechanics. In particular, we obtain the reduced variational principles and the associated Poisson brackets. The special case of higher order Euler-Poincar\'e and Lie-Poisson reduction is also studied in detail.