Unambiguous Formalism for Higher-Order Lagrangian Field Theories

TL;DR

This paper introduces a unique, intrinsic geometric formalism for higher-order field theories that eliminates ambiguities in traditional approaches by unifying Lagrangian and Hamiltonian descriptions using a Skinner-Rusk framework.

Contribution

It develops a differential-geometric, unambiguous formalism for higher-order field theories that provides a global, intrinsic version of the Euler-Lagrange equations, avoiding traditional arbitrariness.

Findings

Provides a unique geometric formulation for higher-order field theories.

Derives a global intrinsic Euler-Lagrange equation.

Illustrates the formalism with several examples.

Abstract

The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher-order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both, the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher-order jet bundle and the canonical multisymplectic form on its dual. As both of these objects are uniquely defined, the Skinner-Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler-Lagrange equations for higher-order field theories. Several examples illustrate our construction.