Higher-order Mechanics: Variational Principles and other topics

TL;DR

This paper reviews the Skinner-Rusk formalism for higher-order dynamical systems and introduces a unified geometric variational principle that derives both Lagrangian and Hamiltonian equations.

Contribution

It presents a unified geometric variational principle for higher-order systems, bridging Lagrangian and Hamiltonian formalisms within a single framework.

Findings

Unified geometric variational principle derived for higher-order systems

Lagrangian and Hamiltonian equations obtained from the unified formalism

Standard formulations recovered from the unified approach

Abstract

After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework.