Tulczyjew Triples in Higher Derivative Field Theory

TL;DR

This paper extends the Tulczyjew triple framework to higher derivative classical field theories, providing a covariant, geometric, and complete formalism that unifies Lagrangian and Hamiltonian approaches.

Contribution

It introduces a novel geometric construction of Tulczyjew triples for arbitrary high order field theories, incorporating affine jet bundle geometry and reduction procedures.

Findings

Derived covariant Tulczyjew triples for high order theories

Unified Lagrangian and Hamiltonian formalisms in a geometric setting

Maintained presymplectic structures in the extended framework

Abstract

The geometrical structure known as Tulczyjew triple has been used with success in analytical mechanics and first order field theory to describe a wide range of physical systems including Lagrangian/Hamiltonian systems with constraints and/or sources, or with singular Lagrangian. Starting from the first principles of the variational calculus we derive Tulczyjew triples for classical field theories of arbitrary high order, i.e. depending on arbitrary high derivatives of the fields. A first triple appears as the result of considering higher order theories as first order ones with configurations being constrained to be holonomic jets. A second triple is obtained after a reduction procedure aimed at getting rid of nonphysical degrees of freedom. This picture we present is fully covariant and complete: it contains both Lagrangian and Hamiltonian formalisms, in particular the Euler-Lagrange equations. Notice that, the required Geometry of jet bundles is affine (as opposed to the linear Geometry of the tangent bundle). Accordingly, the notions of affine duality and affine phase space play a distinguished role in our picture. In particular the Tulczyjew triples in this paper consist of morphisms of double affine-vector bundles which, moreover, preserve suitable presymplectic structures.