Calculus of Variations II

This book by two of the foremost researchers and writers in the field is the first part of a treatise that will cover the subject in breadth and depth, paying special attention to the historical origins, partly in applications, e.g. from geometrical optics, of parts of the theory. A variety of aids to the reader are provided: the detailed table of contents, an introduction to each chapter, section and subsection, plus the references in the Scholia to each chapter, in the (historical) footnotes, and in the biblio- graphy, and finally an index of the examples used throughout the 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.

Inhalt
CALCULUS OF VARIATIONS I - The Lagrangian Formalism: Part I: The First Variation and Necessary Conditions: The First Variation; Variational Problems with Subsidiary Conditions; General Variational Formulas.- Part II: The Second Variation and Sufficient Conditions; Second Variation, Excess Function, Convexity; Weak Minimizers and Jacobi Theory; Weierstrass Field Theory for One-dimensional Integrals and Strong Minimizers. CALCULUS OF VARIATIONS II - The Hamiltonian Formalism: Part III: Canonical Formalism and Hamilton-Jacobi Theory; Legendre Transformation, Hamiltonian Systems, Convexity, Field Theories; Parametric Variational Integrals.- Part IV: Hamilton-Jacobi Theory and Canonical Transformations: Hamilton-Jacobi Theory and Canonical Transformations; Partial Differential Equations of First Order and Contact Transformations.

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