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

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. Preface Contents of Calculus of Variations I and II Introduction Contents of Calculus of Variations I -- The Lagrangian Formalism Contents of Calculus of Variations II -- The Hamiltonian Formalism Part I. The First Variation and Necessary Conditions Chapter 1. The First Variation 1. Critical Points of Functionals 2. Vanishing First Variation and Necessary Conditions 2.1. The First Variation of Variational Integrals 2.2. The Fundamental Lemma of the Calculus of Variations, Euler's Equations, and the Euler Operator L_F 2.3. Mollifiers. Variants of the Fundamental Lemma 2.4. Natural Boundary Conditions 3. Remarks on the Existence and Regularity of Minimizers 3.1. Weak Extremals Which Do Not Satisfy Euler's Equation. A Regularity Theorem for One-Dimensional Variational Problems 3.2. Remarks on the Existence of Minimizers 3.3. Broken Extremals 4. Null Lagrangians 4.1. Basic Properties of Null Lagrangians 4.2. Characterization of Null Lagrangians 5. Variational Problems of Higher Order 6. Scholia Chapter 2. Variational Problems with Subsidiary Conditions 1. Isoperimetric Problems 2. Mappings into Manifolds: Holonomic Constraints 3. Nonholonomic Constraints 4. Constraints at the Boundary. Transversality 5. Scholia Chapter 3. General Variational Formulas 1. Inner Variations and Inner Extremals. Noether Equations 2. Strong Inner Variations, and Strong Inner Extremals 3. A General Variational Formula 4. Emmy Noether's Theorem 5. Transformation of the Euler Operator to New Coordinates 6. Scholia Part II. The Second Variation and Sufficient Conditions Chapter 4. Second Variation, Excess Function, Convexity 1. Necessary Conditions for Relative Minima 1.1. Weak and Strong Minimizers 1.2. Second Variation: Accessory Integral and Accessory Lagrangian 1.3. The Legendre-Hadamard Condition 1.4. The Weierstrass Excess Function E_F and Weierstrass's Necessary Condition 2. Sufficient Conditions for Relative Minima Based on Convexity Arguments 2.1. A Sufficient Condition Based on Definiteness of the Second Variation 2.2. Convex Lagrangians 2.3. The Method of Coordinate Transformations 2.4. Application of Integral Inequalities 2.5. Convexity Modulo Null Lagrangians 2.6. Calibrators 3. Scholia Chapter 5. Weak Minimizers and Jacobi Theory 1. Jacobi Theory: Necessary and Sufficient Conditions for Weak Minimizers Based on Eigenvalue Criteria for the Jacobi Operator 1.1. Remarks on Weak Minimizers 1.2. Accessory Integral and Jacobi Operator 1.3. Necessary and Sufficient Eigenvalue Criteria for Weak Minima 2. Jacobi Theory for One-Dimensional Problems in One Unknown Function 2.1. The Lemmata of Legendre and Jacobi 2.2. Jacobi Fields and Conjugate Values 2.3. Geometric Interpretation of Conjugate Points 2.4. Examples 3. Scholia Chapter 6. Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers 1. The Geometry of One-Dimensional Fields 1.1. Formal Preparations: Fields, Extremal Fields, Mayer Fields and Mayer Bundles, Stigmatic Ray Bundles 1.2. Caratheodory's Royal Road to Field Theory 2. Embedding of Extremals 2.1. Embedding of Regular Extremals into Mayer Fields 2.2. Jacobi's Envelope Theorem 2.3. Catenary and Brachystochrone 2.4. Field-like Mayer Bundles, Focal Points and Caustics 3. Field Theory for Multiple Integrals in the Scalar Case: Lichtenstein's Theorem 4. Scholia Supplement. Some Facts from Differential Geometry and Analysis 1. Euclidean Spaces 2. Some Function Classes 3. Vector and Covector Fields. Transformation Rules 4. Differential Forms 5. Curves in R^N 6. Mean Curvature and Gauss Curvature A List of Examples Length and Geodesics Dirichlet Integral and Harmonic Maps Curvature Functionals Null Lagrangians Counterexamples Mechanics Optics Canonical and Contact Transformations Bibliography Subject Index 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. 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.