Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems

TL;DR

This paper extends the Lagrangian-Hamiltonian unified formalism to higher-order autonomous mechanical systems, providing a comprehensive geometric framework and analyzing physical models within this new approach.

Contribution

It offers the first complete generalization of the formalism for higher-order systems, bridging a gap in the geometric description of complex mechanical models.

Findings

Developed a geometric framework for higher-order systems

Unified Lagrangian and Hamiltonian formalisms in a single approach

Analyzed physical models using the new formalism

Abstract

The Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for first-order and higher-order field theories. However, a complete generalization to higher-order mechanical systems has yet to be described. In this work, after reviewing the natural geometrical setting and the Lagrangian and Hamiltonian formalisms for higher-order autonomous mechanical systems, we develop a complete generalization of the Lagrangian-Hamiltonian unified formalism for these kinds of systems, and we use it to analyze some physical models from this new point of view.