On the geometry of higher-order variational problems on Lie groups

Abstract: In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we deduce an intrinsic framework for this type of dynamical systems. Interesting applications as, for instance, a geometric derivation of the higher-order Euler-Poincar\'e equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.