Calculus of Variations

Description

Cover; Title Page; Copyright Page; Authors' Preface; Translator's Preface; Table of Contents; 1 Elements of the Theory; 1: Functionals. Some Simple Variational Problems; 2: Function Spaces; 3: The Variation of a Functional. A Necessary Condition for an Extremum; 4: The Simplest Variational Problem. Euler'S Equation; 5: The Case of Several Variables; 6: A Simple Variable End Point Problem; 7: The Variational Derivative; 8: Invariance of Euler'S Equation, Problems; 2 Further Generalizations; 9: The Fixed End Point Problem for n Unknown Functions; 10. Variational Problems in Parametric Form.;Based on a series of lectures given by I.M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. The reader who merely wishes to become familiar with the most basic concepts and methods of the calculus of variations need only study the first chapter. Students wishing a more extensive treatment, however, will find the first six chapters comprise a complete university-level course in the subject, including the theory of fields and sufficient conditions for weak and strong extrema. Chapter 7 considers the application of variational methods to the study of systems with infinite degrees of freedom, and Chapter 8 deals with direct methods in the calculus of variations. The problems following each chapter were made especially for this English-language edition, and many of them comment further on corresponding parts of the text. Two appendices and suggestions for supplementary reading round out the text. Substantially revised and corrected by the translator, this inexpensive new edition will be welcomed by advanced undergraduate and graduate students of mathematics and physics. Cover Title Page Copyright Page Authors' Preface Translator's Preface Table of Contents 1 Elements of the Theory 1: Functionals. Some Simple Variational Problems 2: Function Spaces 3: The Variation of a Functional. A Necessary Condition for an Extremum 4: The Simplest Variational Problem. Euler'S Equation 5: The Case of Several Variables 6: A Simple Variable End Point Problem 7: The Variational Derivative 8: Invariance of Euler'S Equation, Problems 2 Further Generalizations 9: The Fixed End Point Problem for n Unknown Functions 10. Variational Problems in Parametric Form. 11: Functionals Depending on Higher-Order Derivatives12: Variational Problems with Subsidiary Conditions, Problems 3 The General Variation of a Functional 13: Derivation of the Basic Formula 14: End Points Lying on Two Given Curves or Surfaces 15: Broken Extremals. The Weierstrass-Erdmann Conditions, Problems 4 The Canonical form of the Euler Equations and Related Topics 16: The Canonical Form of the Euler Equations 17: First Integrals of the Euler Equations 18: The Legendre Transformation 19: Canonical Transformations 20: Noether'S Theorem 21: The Principle of Least Action. 22: Conservation Laws23: The Hamilton-Jacobi Equation. Jacobi'S Theorem, Problems 5 The second Variation. Sufficient Conditions for a weak Extremum 24: Quadratic Functionals. The Second Variation of a Functional 25: The Formula for the Second Variation. Legendre'S Condition 26: Analysis of the Quadratic Functional 27: Jacobi'S Necessary Condition. More on Conjugate Points 28: Sufficient Conditions for a Weak Extremum 29: Generalization to n Unknown Functions 30: Connection Between Jacobi'S Condition and the Theory of Quadratic Forms, Problems. 6 Fields. Sufficient Conditions for a Strong Extremum31: Consistent Boundary Conditions. General Definition of a Field 32: The Field of a Functional 33: Hilbert'S Invariant Integral 34: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum, Problems 7 Variational Problems Involving Multiple Integrals 35: Variation of a Functional Defined on a Fixed Region 36: Variational Derivation of the Equations of Motion of Continuous Mechanical Systems 37: Variation of a Functional Defined on a Variable Region 38: Applications to Field Theory, Problems. 8 Direct Methods in the Calculus of Variations39: Minimizing Sequences 40: The Ritz Method and the Method of Finite Differences 41: The Sturm-Liouville Problem, Problems Appendix I Propagation of Disturbances and the Canonical Equations Appendix II Variational methods in Problems of optimal control Bibliography Index.