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

TL;DR

This paper develops a geometric framework for higher-order variational problems on Lie groups, deriving intrinsic equations and applications like higher-order Euler-Poincaré equations and control systems.

Contribution

It introduces a novel geometric setting using left-trivialization and Skinner-Rusk formalism for higher-order Lagrangian problems on Lie groups.

Findings

Derived higher-order Euler-Poincaré equations geometrically.

Applied the framework to optimal control of underactuated systems.

Provided an intrinsic approach to higher-order dynamical systems 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.