Calculus of Variations II

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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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.

• Calculus of Variations
• Convexity
• Hamiltonian Formalism
• Lagrangian Formalism
• differential equation

• Scuola Normale Superiore, Pisa, Italy

  Mariano Giaquinta

• Mathematisches Institut, Universität Bonn, Bonn, Germany

  Stefan Hildebrandt

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