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An exterior algebraic derivation of the euler–lagrange equations from the principle of stationary action

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dc.contributor.author Colombaro, Ivano
dc.contributor.author Font Segura, Josep
dc.contributor.author Martínez, Alfonso, 1973-
dc.date.accessioned 2022-06-28T06:13:06Z
dc.date.available 2022-06-28T06:13:06Z
dc.date.issued 2021
dc.identifier.citation Colombaro I, Font-Segura J, Martínez A. An exterior algebraic derivation of the euler–lagrange equations from the principle of stationary action. Mathematics. 2021;9(18):2178. DOI: 10.3390/math9182178
dc.identifier.issn 2227-7390
dc.identifier.uri http://hdl.handle.net/10230/53606
dc.description.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.
dc.description.sponsorship This work was funded in part by the Spanish Ministry of Science, Innovation and Universities under grants TEC2016-78434-C3-1-R and BES-2017-081360.
dc.format.mimetype application/pdf
dc.language.iso eng
dc.publisher MDPI
dc.relation.ispartof Mathematics. 2021;9(18):2178.
dc.rights © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
dc.rights.uri https://creativecommons.org/licenses/by/4.0/
dc.title An exterior algebraic derivation of the euler–lagrange equations from the principle of stationary action
dc.type info:eu-repo/semantics/article
dc.identifier.doi http://doi.org/10.3390/math9182178
dc.subject.keyword Euler–Lagrange equations
dc.subject.keyword exterior algebra
dc.subject.keyword exterior calculus
dc.subject.keyword tensor calculus
dc.subject.keyword action principle
dc.subject.keyword Lagrangian
dc.subject.keyword electromagnetism
dc.subject.keyword Maxwell equations
dc.relation.projectID info:eu-repo/grantAgreement/ES/1PE/TEC2016-784
dc.relation.projectID info:eu-repo/grantAgreement/ES/2PE/BES-2017-081360
dc.rights.accessRights info:eu-repo/semantics/openAccess
dc.type.version info:eu-repo/semantics/publishedVersion

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