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Calculus of Variations II

  • Book
  • © 2004

Overview

Authors:
  1. Mariano Giaquinta

    1. Scuola Normale Superiore, Pisa, Italy

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  2. Stefan Hildebrandt

    1. Mathematisches Institut, Universität Bonn, Bonn, Germany

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 311)

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Table of contents (4 chapters)

  1. Front Matter

    Pages I-XXIX
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  2. Canonical Formalism and Parametric Variational Problems

    1. Front Matter

      Pages 1-1
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    2. Legendre Transformation, Hamiltonian Systems, Convexity, Field Theories

      • Mariano Giaquinta, Stefan Hildebrandt
      Pages 3-152

    3. Parametric Variational Integrals

      • Mariano Giaquinta, Stefan Hildebrandt
      Pages 153-280

  3. Hamilton-Jacobi Theory and Partial Differential Equations of First Order

    1. Front Matter

      Pages 281-281
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    2. Hamilton-Jacobi Theory and Canonical Transformations

      • Mariano Giaquinta, Stefan Hildebrandt
      Pages 283-440

    3. Partial Differential Equations of First Order and Contact Transformations

      • Mariano Giaquinta, Stefan Hildebrandt
      Pages 441-604

  4. Back Matter

    Pages 605-652
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Keywords

About this book

This book describes the classical aspects of the variational calculus which are of interest to analysts, geometers and physicists alike. Volume 1 deals with the for­ mal apparatus of the variational calculus and with nonparametric field theory, whereas Volume 2 treats parametric variational problems as weIl as Hamilton­ Jacobi theory and the classical theory of partial differential equations of first order. In a subsequent treatise we shall describe developments arising from Hilbert's 19th and 20th problems, especially direct methods and regularity theory. Of the classical variational calculus we have particularly emphasized the often neglected theory of inner variations, i. e. of variations of the independent variables, which is a source of useful information such as monotonicity for­ mulas, conformality relations and conservation laws. The combined variation of dependent and independent variables leads to the general conservation laws of Emmy Noether, an important tool in exploitingsymmetries. Other parts of this volume deal with Legendre-Jacobi theory and with field theories. In particular we give a detailed presentation of one-dimensional field theory for non para­ metric and parametric integrals and its relations to Hamilton-Jacobi theory, geometrieal optics and point mechanics. Moreover we discuss various ways of exploiting the notion of convexity in the calculus of variations, and field theory is certainly the most subtle method to make use of convexity. We also stress the usefulness of the concept of a null Lagrangian which plays an important role in several instances.

Authors and Affiliations

  • Scuola Normale Superiore, Pisa, Italy

    Mariano Giaquinta

  • Mathematisches Institut, Universität Bonn, Bonn, Germany

    Stefan Hildebrandt

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