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Lagrange Multipliers in Infinite Dimensional Spaces, Examples of Application

Encyclopedia of Continuum Mechanics

Synonyms

Infinite-dimensional constrained mechanical systems

Definitions

The Lagrange multipliers method is used in mathematical analysis, in mechanics, in economics, and in several other fields, to deal with the search of the global maximum or minimum of a function, in the presence of a constraint. The usual technique, applied to the case of finite-dimensional systems, transforms the constrained optimization problem into an unconstrained one, by means of the introduction of one or more multipliers and of a suitable Lagrangian function, to be optimized. In mechanics, several optimization problems can be applied to infinite-dimensional systems. Lagrange multipliers method can be applied also to these cases.

Introduction

In this entry we show that the theorem of Lagrange multipliers in infinite-dimensional systems (dell’Isola and Di Cosmo, 2018) can be a very powerful tool for dealing with constrained problems also in infinite-dimensional spaces. This tool is powerful but must be used...

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References

  • Dautray R, Lions J-L (2012) Mathematical analysis and numerical methods for science and technology: volume 3 spectral theory and applications. Springer Science Business Media, Springer-Verlag Berlin Heidelberg

    MATH  Google Scholar 

  • Della Corte A, dell’Isola F, Seppecher P (2015) The postulations à la D‘Alembert and à la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc R Soc A Math Phys Eng Sci 471(2183):20150415

    Google Scholar 

  • dell’Isola F, Di Cosmo F (2018) Lagrange multipliers in infinite-dimensional systems, methods of. In: Altenbach H, Öchsner A (eds) Encyclopedia of continuum mechanics. Springer, Berlin/Heidelberg

    Google Scholar 

  • dell’Isola F, Madeo A, Seppecher P (2012) How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D‘Alembert”. Z Angew Math Phys 63(6):1119–1141

    Article  MathSciNet  Google Scholar 

  • dell’Isola F, Madeo A, Seppecher P (2016) Cauchy tetrahedron argument applied to higher contact interactions. Arch Ration Mech Anal 219(3):1305–1341

    Article  MathSciNet  Google Scholar 

  • Forest S, Cordero NM, Busso EP (2011) First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput Mater Sci 50(4):1299–1304

    Article  Google Scholar 

  • Germain P (1973) La méthode des puissances virtuelles en mécanique des milieux continus. J Mécanique 12:236–274

    MATH  Google Scholar 

  • Glüge R (2018) Continuum mechanics basics, introduction and notations. In: Altenbach H, Öchsner A (eds) Encyclopedia of continuum mechanics. Springer, Berlin/Heidelberg

    Google Scholar 

  • Lagrange JL (1853) Mécanique analytique, vol 1. Mallet-Bachelier, Paris

    Google Scholar 

  • Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78

    Article  MathSciNet  Google Scholar 

  • Rudin W (1987) Real and complex analysis. McGraw–Hill, New York

    MATH  Google Scholar 

  • Schwartz L (1957) Théorie des distributions, vol 2. Hermann, Paris

    MATH  Google Scholar 

  • Schweizer B (2018) On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. In: Rocca E, Stefanelli U, Truskinovsky L, Visintin A (eds) Trends in applications of mathematics to mechanics. Springer INdAM series, Springer Cham, Vol 21, pp 65–79

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Sapienza University, Rome, Italy

    A. Bersani

  2. Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza”, Rome, Italy

    F. dell’Isola

  3. Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, Università degli Studi dell’Aquila, L’Aquila, Italy

    F. dell’Isola

  4. International Research Center for the Mathematics and Mechanics of Complex Systems, L’Aquila, Italy

    F. dell’Isola

  5. Institut de Mathématiques, Université de Toulon, Toulon, France

    P. Seppecher

Authors
  1. A. Bersani
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  2. F. dell’Isola
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  3. P. Seppecher
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Corresponding author

Correspondence to F. dell’Isola .

Editor information

Editors and Affiliations

  1. Otto-von-Guericke University Magdeburg, Chair of Engineering Mechanics, Institut, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. Fakultät für Maschinenbau, Hochschule Esslingen, Esslingen am Neckar, Baden-Württemberg, Germany

    Andreas Öchsner

Section Editor information

  1. Dipartimento di Ingegneria Strutturale e Geotecnica, Università degli Studi di Roma “La Sapienza”, Rome, Italy

    Francesco dell’Isola

  2. Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale, Università degli Studi dell’Aquila, L’Aquila, Italy

    Francesco dell’Isola

  3. International Research Center for the Mathematics and Mechanics of Complex Systems, L’Aquila, Italy

    Francesco dell’Isola

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Bersani, A., dell’Isola, F., Seppecher, P. (2019). Lagrange Multipliers in Infinite Dimensional Spaces, Examples of Application. In: Altenbach, H., Öchsner, A. (eds) Encyclopedia of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53605-6_266-1

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  • DOI: https://doi.org/10.1007/978-3-662-53605-6_266-1

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53605-6

  • Online ISBN: 978-3-662-53605-6

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

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Chapter history

  1. Latest

    Lagrange Multipliers in Infinite Dimensional Spaces, Examples of Application
    Published:
    25 October 2019

    DOI: https://doi.org/10.1007/978-3-662-53605-6_266-2

  2. Original

    Lagrange Multipliers in Infinite Dimensional Spaces, Examples of Application
    Published:
    24 September 2019

    DOI: https://doi.org/10.1007/978-3-662-53605-6_266-1

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