Classical field theories of first order and lagrangian submanifolds of premultisymplectic manifolds

TL;DR

This paper presents a geometric framework for classical first-order field theories using Lagrangian submanifolds of premultisymplectic manifolds, linking Euler-Lagrange and Hamiltonian formalisms through Tulczyjew's triple.

Contribution

It introduces a novel geometric approach connecting Lagrangian submanifolds with classical field equations via a specialized Tulczyjew's triple for premultisymplectic manifolds.

Findings

Euler-Lagrange equations are characterized as Lagrangian submanifolds.

Hamilton-De Donder-Weyl equations are also described as Lagrangian submanifolds.

The framework unifies Lagrangian and Hamiltonian formalisms geometrically.

Abstract

A description of classical field theories of first order in terms of Lagrangian submanifolds of premultisymplectic manifolds is presented. For this purpose, a Tulczyjew's triple associated with a fibration is discussed. The triple is adapted to the extended Hamiltonian formalism. Using this triple, we prove that Euler-Lagrange and Hamilton-De Donder-Weyl equations are the local equations defining Lagrangian submanifolds of a premultisymplectic manifold.