An Exterior Algebraic Derivation of the Euler-Lagrange Equations from the Principle of Stationary Action

TL;DR

This paper introduces a coordinate-free, exterior algebra-based approach to deriving the Euler-Lagrange equations in field theory, providing new forms that simplify the equations and applying this to generalized electromagnetic fields.

Contribution

It presents two novel coordinate-free forms of the Euler-Lagrange equations using exterior algebra, extending the derivation to multivector fields and their derivatives.

Findings

Derived coordinate-free Euler-Lagrange equations using exterior algebra.

Applied the approach to generalized electromagnetic fields of arbitrary grade.

Obtained Maxwell equations from the Lagrangian density via vector derivatives.

Abstract

In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler-Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial derivation of these equations for field theories, that gives separate equations for the field components, two related coordinate-free forms of the Euler-Lagrange equations are derived. These alternative forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian density that only depends on the first-order derivatives of the field. The second form, expressed in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have, by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as Euler--Lagrange equations.