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Calculus of Variations and Partial Differential Equations of First Order: Second Edition
C. Carathéodory
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AMS Chelsea Publishing
1965; 402 pp; hardcover
Reprint/Revision History:
second revised English printing 1982; second AMS printing 2001
ISBN-10: 0-8218-1999-2
ISBN-13: 978-0-8218-1999-9
List Price: US$51
Member Price: US$45.90
Order Code: CHEL/318.H
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In this second English edition of Carathéodory's famous work (originally published in German), the two volumes of the first edition have been combined into one (with a combination of the two indexes into a single index). There is a deep and fundamental relationship between the differential equations that occur in the calculus of variations and partial differential equations of the first order: in particular, to each such partial differential equation there correspond variational problems. This basic fact forms the rationale for Carathéodory's masterpiece. Includes a Guide to the Literature and an Index.

From the Preface: "The book consists of two parts. In the first part, I have made an attempt to simplify the presentation of the theory of partial differential equations to the first order so that its study will require little time and also be accessible to the average student of mathematics ... The second part, which contains the Calculus of Variations, can also be read independently if one refers back to earlier sections in Part I ... I have never lost sight of the fact that the Calculus of Variations, as it is presented in Part II, should above all be a servant of Mechanics. Therefore, I have in particular prepared everything from the very outset for treatment in multidimensional spaces."

Table of Contents

  • Continuous convergence, implicit functions, ordinary differential equations
  • Fields of curves and multidimensional surfaces, complete systems
  • Partial differential equations of the first order, theory of characteristics
  • Poisson brackets, systems of partial differential equations of the first order
  • Elements of tensor calculus
  • Canonical transformations
  • Contact transformations
  • The Pfaff problem
  • Function groups
  • The integration theories of Lagrange, Jacobi, Adolph Mayer and Lie
  • Ordinary maxima and minima. Quadratic forms
  • Simple variational problems in the small
  • Variational problems in parametric representation
  • Positive-definite variational problems
  • Quadratic variational problems. Theory of the second variation
  • The boundary-value problem and the question of the absolute minimum
  • Closed extremals. Periodic variational problems
  • The problem of Lagrange
  • Guide to the literature
  • Index
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