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-Poincaré equations, optimal control of underactuated control systems whose configuration space is a Lie group are shown, among others, along the paper.

20 pages, 4 figures

References in corpus (3)

• Invariant higher-order variational problems
• Invariant higher-order variational problems II
• Some applications of quasi-velocities in optimal control

Cited by in corpus (11)

• Higher-order discrete variational problems with constraints
• Tulczyjew's Triplet for Lie Groups II: Dynamics
• Geometric integrators for higher-order mechanics on Lie groups
• Inexact trajectory planning and inverse problems in the Hamilton--Pontryagin framework
• Reductions of Topologically Massive Gravity I: Hamiltonian Analysis of The Second Order Degenerate Lagrangians
• Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions
• Lagrange-Poincaré reduction for optimal control of underactuated mechanical systems
• Reductions of Dynamics on Second Iterated Bundles of Lie Groups
• Higher-Order Analogs of Lie Algebroids via Vector Bundle Comorphisms
• Variational integrators for underactuated mechanical control systems with symmetries
• Geometrical structures of higher-order dynamical systems and field theories