Calculus of Variations I (Grundlehren der mathematischen Wissenschaften (310))

This 2-volume treatise by two of the leading researchers and writers in the field, quickly established itself as a standard reference. It pays special attention to the historical aspects and the origins partly in applied problems - such as those of geometric optics - of parts of the theory. A variety of aids to the reader are provided, beginning with the detailed table of contents, and including an introduction to each chapter and each section and subsection, an overview of the relevant literature (in Volume II) besides the references in the Scholia to each chapter in the (historical) footnotes, and in the bibliography, and finally an index of the examples used through out 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 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 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. Front Matter....Pages I-XXIX Front Matter....Pages 1-1 The First Variation....Pages 3-86 Variational Problems with Subsidiary Conditions....Pages 87-144 General Variational Formulas....Pages 145-214 Front Matter....Pages 215-215 Second Variation, Excess Function, Convexity....Pages 217-263 Weak Minimizers and Jacobi Theory....Pages 264-309 Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers....Pages 310-399 Back Matter....Pages 400-474

This two-volume treatise is a standard reference in the field. It pays special attention to the historical aspects and the origins partly in applied problems-such as those of geometric optics-of parts of the theory. It contains an introduction to each chapter, section, and subsection and an overview of the relevant literature in the footnotes and bibliography. It also includes an index of the examples used throughout the book.