Ebook: Calculus of Variations

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.