Calculus of Variations I

1. Front Matter

   Pages I-XXIX

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2. The First Variation and Necessary Conditions

   1. Front Matter

      Pages 1-1

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   2. The First Variation

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 3-86

   3. Variational Problems with Subsidiary Conditions

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 87-144

   4. General Variational Formulas

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 145-214

3. The Second Variation and Sufficient Conditions

   1. Front Matter

      Pages 215-215

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   2. Second Variation, Excess Function, Convexity

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 217-263

   3. Weak Minimizers and Jacobi Theory

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 264-309

   4. Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers

      • Mariano Giaquinta, Stefan Hildebrandt

      Pages 310-399

4. Back Matter

   Pages 400-474

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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 well as Hamilton­ Jacobi theory and the classical theory of partial differential equations of first ordel;. 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 mono tonicity 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 exploiting symmetries. 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 nonpara­ metric and parametric integrals and its relations to Hamilton-Jacobi theory, geometrical 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 we give an exposition of Hamilton-Jacobi several instances.

• Calculus of Variations
• Convexity
• Gaussian curvature
• Hamiltonian Formalism
• Lagrangian Formalism
• Mean curvature
• Vector field
• calculus
• curvature
• differential equation
• differential geometry

• Scuola Normale Superiore, Pisa, Italy

  Mariano Giaquinta

• Mathematisches Institut, Universität Bonn, Bonn, Germany

  Stefan Hildebrandt

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