Unified formalism for higher-order non-autonomous dynamical systems

TL;DR

This paper develops a comprehensive geometric framework for higher-order non-autonomous mechanical systems, extending existing formalisms and deriving key equations for both regular and singular cases.

Contribution

It generalizes the Skinner-Rusk formalism to higher-order non-autonomous systems and derives the Lagrangian and Hamiltonian formalisms within this unified approach.

Findings

Extended Skinner-Rusk formalism for higher-order non-autonomous systems

Derived Legendre-Ostrogradsky map and Euler-Lagrange equations

Applied framework to specific regular and singular physical systems

Abstract

This work is devoted to giving a geometric framework for describing higher-order non-autonomous mechanical systems. The starting point is to extend the Lagrangian-Hamiltonian unified formalism of Skinner and Rusk for these kinds of systems, generalizing previous developments for higher-order autonomous mechanical systems and first-order non-autonomous mechanical systems. Then, we use this unified formulation to derive the standard Lagrangian and Hamiltonian formalisms, including the Legendre-Ostrogradsky map and the Euler-Lagrange and the Hamilton equations, both for regular and singular systems. As applications of our model, two examples of regular and singular physical systems are studied.