Lagrangian-Hamiltonian unified formalism for field theory

TL;DR

This paper extends the Rusk-Skinner unified formalism from mechanics to first-order classical field theories, integrating Lagrangian and Hamiltonian descriptions into a single geometric framework.

Contribution

It introduces a unified geometric formalism for first-order classical field theories that encompasses both regular and singular cases, advancing the theoretical foundation for future control applications.

Findings

Unified formalism captures key features of Lagrangian and Hamiltonian approaches.

Applicable to both regular and singular field theories.

Lays groundwork for optimal control of PDEs.

Abstract

The Rusk-Skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems. It incorporates all the characteristics of Lagrangian and Hamiltonian descriptions of these systems (including dynamical equations and solutions, constraints, Legendre map, evolution operators, equivalence, etc.). In this work we extend this unified framework to first-order classical field theories, and show how this description comprises the main features of the Lagrangian and Hamiltonian formalisms, both for the regular and singular cases. This formulation is a first step toward further applications in optimal control theory for PDE's.