Calculus of Variations
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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 specially 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.
- ELEMENTS OF THE THEORY
- Functionals. Some Simple Variational Problems
- Function Spaces
- The Variation of a Functional
- A Necessary Condition for an Extremum
- The Simplest Variational Problem. Euler's Equaiton
- The Case of Several Variables
- A Simple Variable End Point Problem
- The Variational Derivative
- Invariance of Euler's Equaiton
- Problems
- FURTHER GENERALIZATIONS
- The Fixed End Point Problem for n Unknown Functions
- Variational Problems in Parametric Form
- Functionals Depending on Higher-Order Derivatives
- Variational Problems with Subsidiary Conditions
- Problems
- THE GENERAL VARIATION OF A FUNCTIONAL
- Derivation of the Basic Formula
- End Points Lying on Two Given Curves or Surfaces
- Broken Extremals
- The Weierstrass-Erdmann Conditions
- Problems
- THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS
- THE SECOND VARIATION. SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM
- Quadratic Functionals. The Second Variation of a Functional
- The Formula for the Second Variation
- Legendre's Condition
- Analysis of the Quadratic Funtional
- Jacobi's Necessary Condition. More on Conjugade Points
- Sufficient Conditions for a Weak Extremum
- Generalization to n Unknown Functions
- Connection Between Jacobi's Condition and the Theory of Quadratic Forms
- Problems
- FIELDS. SUFFICIENT CONDITIONS FOR A STRONG EXTREMUM
- Consistent Boundary Conditions. General Definition of a Field
- The Field of a Functional
- Hilbert's Invariant Integral
- The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum
- Problems
- VARIATIONAL PROBLEMS INVOLVING MULTIPLE INTEGRALS
- Variation of a Functional Defined on a Fixed Region
- Variational Derivation of the Equations of Motion of Continuous Mechanical Systems
- Variation of a Functional Defined on a Variable Region
- Applications to Field Theory
- Problems
- DIRECT METHODS IN THE CALCULUS OF VARIATIONS
- Minimizing Sequences
- The Ritz Method and the Method of Finite Differences
- The Sturm-Liouville Problem
- Problems
- (Appendix I) PROPAGATION OF DISTURBANCES AND THE CANONICAL EQUATIONS
- (Appendix II) VANIATIONAL METHODS IN PROBLEMS OF OPTIMAL CONTROL
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