Reduced variational formulations in free boundary continuum mechanics
type: Item
instance of: scholarly article
title: Reduced variational formulations in free boundary continuum mechanics
zbMATH Open document ID: 1260.37031
author: François Gay-Balmaz
author: J. E. Marsden
author: T. S. Ratiu
published in: Journal of Nonlinear Science
publication date: 2012-10-29T00:00:00Z
full work available at URL: 332 2012 Article 9143.pdf
review text: In free-boundary continuum mechanics, the authors apply the Lagrangian reduction approach, allowing a better use of the material and spatial symmetries of the studied systems. The results of the article are achieved by implementing the Euler-Poincaré reduction for a semidirect product involving the diffeomorphism group of the fluids domain in compressible hydrodynamics. Here both the constrained variational principle and the equations of motion are presented. But the Euler-Poincaré reduction theory is not applied in nonlinear elasticity problems, where the constrained variational principles together with the equations of motion are deduced from the material representation, both in convective and spatial formulation. The same approach of Lagrangian reduction is carried out for free-boundary compressible fluids with surface tension allowing to deduce the equations of motion and their associated constrained variational principles in convective and spatial representation. Lagrangian reduction is also acceptable for general continua, in particularl for fluid-solid interaction problems. Here the extension of the notions of material and spatial symmetries allow to deduce the constrained variational principles in pure spatial or convective variables and to derive the equations of motion in convective and spatial representations.
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 37K65
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 37N15
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 37N10
Mathematics Subject Classification ID: 74F10
Mathematics Subject Classification ID: 76N99
zbMATH DE Number: 6100219
zbMATH Keywords: symmetry
zbMATH Keywords: reduction
zbMATH Keywords: variational principle
zbMATH Keywords: constraints
zbMATH Keywords: material representation
zbMATH Keywords: spatial representation
zbMATH Keywords: convective representation
zbMATH Keywords: Lagrangian
zbMATH Keywords: rigid body
zbMATH Keywords: ideal fluid
zbMATH Keywords: elastic material
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
OpenAlex ID: W2016478498
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DOI: 10.1007/S00332-012-9143-4
description: scientific article; zbMATH DE number 6100219