## Calculus Of Variations – Gelfand, Fomin

In this post, we will see the book Calculus Of Variations by I.M. Gelfand; S.V. Fomin.

This book is a modern introduction to the calculus of variations and certain of its ramifications, and I trust that its fresh and lively point of view will serve to make it a welcome addition to the English-language literature on the subject. The present edition is rather different from the Russian original. With the authors’ consent, I have given free rein to the tendency of any mathematically educated translator to assume the functions of annotator and stylist.

The problems appearing at the end of each of the eight chapters and two appendices were made specifically for the English edition, and many of them comment further on the corresponding parts of the text.

The book was translated from Russian by Richard Silverman and was published in 1963.

You can get the book here.

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# Contents

## 1. ELEMENTS OF THE THEORY 1.

1: Functionals. Some Simple Variational Problems, 1.
2: Function Spaces, 4.
3: The Variation of a Functional. A Necessary Condition for an Extremum, 8.
4: The Simplest Variational Problem. Euler’s Equation, 14.
5: The Case of Several Variables, 22.
6: A Simple Variable End Point Problem, 25.
7: The Variational Derivative, 27.
8: Invariance of Euler’s Equation, 29. Problems, 31.

## 2. FURTHER GENERALIZATIONS 34.

9: The Fixed End Point Problem for n Unknown Functions, 34.
10. Variational Problems in Parametric Form, 38.
11: Functionals Depending on Higher-Order Derivatives, 40.
12: Variational Problems with Subsidiary Conditions, 42.
Problems, 50.

## 3. THE GENERAL VARIATION OF A FUNCTIONAL 54.

13: Derivation of the Basic Formula, 54.
14: End Points Lying on Two Given Curves or Surfaces, 59.
15: Broken Extremals. The Weierstrass-Erdmann Conditions, 61.
Problems, 63.

## 4. THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS 67.

16: The Canonical Form of the Euler Equations, 67.
17: First Integrals of the Euler Equations, 70.
18: The Legendre Transformation, 71.
19: Canonical Transformations, 77. 20: Noether’s Theorem, 79.
21: The Principle of Least Action, 83.
22: Conservation Laws, 85.
23: The Hamilton-Jacobi Equation. Jacobi’s Theorem, 88.
Problems, 94,

## 5. THE SECOND VARIATION. SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM 97.

24: Quadratic Functionals. The Second Variation of a Func-tional, 97.
25: The Formula for the Second Variation. Legendre’s Condition, 101. 26: Analysis of the Quadratic Functional | (Ph’? + Qh?) dx, 105.
27: Jacobi’s Necessary Condition. More on Conjugate Points, 111.
28: Sufficient Conditions for a Weak Extremum, 115.
29: Generalization to n Unknown Functions, 117.
30: Connection Between Jacobi’s Condition and the Theory of Quadratic Forms, 125.
Problems, 129.

## 6. FIELDS. SUFFICIENT CONDITIONS FOR A STRONG EXTREMUM 131.

31: Consistent Boundary Conditions. General Definition of a Field, 131.
32: The Field of a Functional, 137.
33: Hilbert’s Invariant Integral, 145.
34: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum, 146.
Problems, 150.

## 7. VARIATIONAL PROBLEMS INVOLVING MULTIPLE INTEGRALS 152.

35: Variation of a Functional Defined on a Fixed Region, 152.
36: Variational Derivation of the Equations of Motion of Continuous Mechanical Systems, 154.
37: Variation of a Functional
Defined on a Variable Region, 168.
38: Applications to Field Theory, 180.
Problems, 190.

## 8. DIRECT METHODS IN THE CALCULUS OF VARIATIONS 192.

39: Minimizing Sequences, 193.
40: The Ritz Method and the Method of Finite Differences, 195.
41: The Sturm-Liouville Problem, 198.
Problems, 206.

## APPENDIX II VARIATIONAL METHODS IN PROBLEMS OF OPTIMAL CONTROL 218.

BIBLIOGRAPHY 227.

INDEX 228.