Line–shaped objects and their balances related to gauge symmetries in continuum theories

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Scholle Markus and Anthony Karl–Heinz 2004Line–shaped objects and their balances related to gauge symmetries in continuum theoriesProc. R. Soc. Lond. A.460875–896http://doi.org/10.1098/rspa.2003.1198SectionRestricted accessResearch articleLine–shaped objects and their balances related to gauge symmetries in continuum theories Markus Scholle Markus Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany () Google Scholar Find this author on PubMed Search for more papers by this author and Karl–Heinz Anthony Karl–Heinz Anthony FB6—Theoretische Physik , Universität–GH Paderborn, Warburger Straβe 100, 33095 Paderborn, Germany () Google Scholar Find this author on PubMed Search for more papers by this author Markus Scholle Markus Scholle Lehrstuhl für Technische Mechanik und Strömungsmechanik, University of Bayreuth, Universitätsstraβe 30, 95440 Bayreuth, Germany () Google Scholar Find this author on PubMed Search for more papers by this author and Karl–Heinz Anthony Karl–Heinz Anthony FB6—Theoretische Physik , Universität–GH Paderborn, Warburger Straβe 100, 33095 Paderborn, Germany () Google Scholar Find this author on PubMed Search for more papers by this author Published:08 March 2004https://doi.org/10.1098/rspa.2003.1198AbstractWithin the Lagrange formalism Noether's theorem is a well–known tool for connecting symmetries of a physical system with homogeneous balance equations. In the context of this paper we call them balance equations of the volume type. They are associated with symmetry groups of the Lie type. However, in physics there are a lot of different balance equations which we call balance equations of the area type. Physically, they are associated with the dynamics of line–shaped objects. In this paper a general theory is presented which supplements Noether's theorem in so far as the area–type balances are associated with regauging symmetry groups of the non–Lie type. The theory is demonstrated for three prominent examples: for the Helmholtz laws of the vortex dynamics in an ideal fluid, for the dislocation dynamics in the dynamical eigenstress problem of elastic crystals, and for the homogeneous Maxwell equations. The theory will also be a valuable tool for solving inverse variational problems, i.e. to construct Lagrangians for physical systems. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Scholle M and Marner F (2016) A generalized Clebsch transformation leading to a first integral of Navier–Stokes equations, Physics Letters A, 10.1016/j.physleta.2016.07.066, 380:40, (3258-3261), Online publication date: 1-Sep-2016. Scholle M and Marner F (2015) The Clebsch transformation and its capabilities towards fluid and solid mechanics, PAMM, 10.1002/pamm.201510232, 15:1, (483-484), Online publication date: 1-Oct-2015. Prakash J, Lavrenteva O and Nir A (2014) Application of Clebsch variables to fluid-body interaction in presence of non-uniform vorticity, Physics of Fluids, 10.1063/1.4891198, 26:7, (077102), Online publication date: 1-Jul-2014. Scholle M, Heining C and Haas A (2008) Comment on: “Variational principle for two-dimensional incompressible inviscid flow” [Phys. Lett. A 371 (2007) 39], Physics Letters A, 10.1016/j.physleta.2008.07.015, 372:36, (5857), Online publication date: 1-Sep-2008. Scholle M (2004) Construction of Lagrangians in continuum theories, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460:2051, (3241-3260), Online publication date: 8-Nov-2004. This Issue08 March 2004Volume 460Issue 2043 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1198Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/03/2004Published in print08/03/2004 License: Citations and impact KeywordsVolume–Type And Area–Type BalancesDislocationsMaxwell's EquationsLagrange FormalismLie And Non–Lie GroupsVortices