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Title: Conjugate flow action functionals

Abstract

We present a new general framework to construct an action functional for a non-potential field theory. The key idea relies on representing the governing equations relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field. Such flow, which will be called the conjugate flow, evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the Gâteaux derivative of the field equations with respect to suitable local bilinear forms. This is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a Lie group manifold. By using a general operator framework, we obtain the determining equations of such manifold and the corresponding conjugate flow action functional. In particular, we study scalar and vector field theories governed by second-order nonlinear partial differential equations. The identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines.

Authors:
 [1]
  1. Division of Applied Mathematics, Brown University, Rhode Island 02912 (United States)
Publication Date:
OSTI Identifier:
22251914
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 54; Journal Issue: 11; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSICAL MECHANICS; CONSERVATION LAWS; CURVILINEAR COORDINATES; FIELD EQUATIONS; FIELD THEORIES; FLUID FLOW; FLUID MECHANICS; LIE GROUPS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS; VECTOR FIELDS

Citation Formats

Venturi, Daniele, E-mail: daniele-venturi@brown.edu. Conjugate flow action functionals. United States: N. p., 2013. Web. doi:10.1063/1.4827679.
Venturi, Daniele, E-mail: daniele-venturi@brown.edu. Conjugate flow action functionals. United States. doi:10.1063/1.4827679.
Venturi, Daniele, E-mail: daniele-venturi@brown.edu. 2013. "Conjugate flow action functionals". United States. doi:10.1063/1.4827679.
@article{osti_22251914,
title = {Conjugate flow action functionals},
author = {Venturi, Daniele, E-mail: daniele-venturi@brown.edu},
abstractNote = {We present a new general framework to construct an action functional for a non-potential field theory. The key idea relies on representing the governing equations relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field. Such flow, which will be called the conjugate flow, evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the Gâteaux derivative of the field equations with respect to suitable local bilinear forms. This is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a Lie group manifold. By using a general operator framework, we obtain the determining equations of such manifold and the corresponding conjugate flow action functional. In particular, we study scalar and vector field theories governed by second-order nonlinear partial differential equations. The identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines.},
doi = {10.1063/1.4827679},
journal = {Journal of Mathematical Physics},
number = 11,
volume = 54,
place = {United States},
year = 2013,
month =
}
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