An Introduction to the Calculus of Variations

PREFACE

CHAPTER I. Introduction

1.1. The fundamental problem. 1.2. Notation. 1.3. Conditions assumed in the fundamental problem. 1.4. The geodesic problem. 1.5. Absolute minima. 1.6. The rounding argument. 1.7. Relative minima: strong and weak variations. 1.8. Minima, maxima, stationary values. 1.9. Historical. 1.10. Extensions of the theory.

CHAPTER II. Fundamental theory

2.1. Two lemmas. 2.2. Necessary conditions for a weak relative minimum, the elementary theory. 2.3. Necessary conditions for a weak relative minimum. 2.4. Euler’s equation. 2.5. The first corner condition. 2.6. The second corner condition. 2.7. Regular arcs, existence and continuity of φ''(x). 2.8. The solutions of Euler’s equation. 2.9. Some special cases. 2.10. Second-order conditions and the search for sufficient conditions. 2.11. Legendre’s necessary condition. 2.12. Non-contemporaneous variations. 2.13. An outline of the theory of the second variation.

CHAPTER III. illustrative examples

3.1. Application of the theory to concrete problems. 3.2. Line integral. 3.3. The integrand a function of y'. 3.4. Hamilton’s principle. 3.5. Weierstrass’s problem. 3.6. The integral ∫ yⁿ ds, genesis of the problem. 3.7. The integral ∫ yⁿ ds for n = –1 and for n = –1/2. 3.8. The integral ∫ √y ds. 3.9. The integral ∫ y ds. 3.10. The integral ∫ √(y(1– y'²)) dx. 3.11. Polar coordinates. 3.12. The Newtonian orbit.

CHAPTER IV. Variable end-points

4.1. Variation from a minimizing curve. 4.2. The variable end-point theorem. 4.3. The problem of the minimizing curve with variable end-points. 4.4. Weierstrass’s necessary condition.

CHAPTER V. The fundamental sufficiency theorem

5.1. Fields of extremals. 5.2. Hilbert’s invariant integral. 5.3. The Pfaffian form u dx + v dy. 5.4. Jacobi’s necessary condition. 5.5. The fundamental sufficiency theorem. 5.6. A notation for the various conditions. 5.7. Problems in which the integrand is a function of y'. 5.8. The integral ∫ f(x, y) ds. 5.9. Problems arising from Hamilton’s principle. 5.10. The integral ∫ (1/2 y'² + cy) dx. 5.11. The integral ∫ (1/2 y'² – y²) dx. 5.12. The integral ∫ (1/2 y'² + 1/y) dx. 5.13. The integral ∫ y2 (1 + y'²) dx. 5.14. The integral ∫ √(y(1–y'²)) dx. 5.15. The construction of a field in the neighbourhood of a given extremal. 5.16. Sufficient conditions for a weak relative minimum. 5.17. The absolute minimum for ∫ √y ds. 5.18. Calculation of the integrals. 5.19. Variable end-points.

CHAPTER VI. The isoperimetrical problem

6.1. Steiner’s problem. 6.2. The classical isoperimetrical problem. 6.3. A “physical” solution of the classical isoperimetrical problem. 6.4. The isoperimetrical problem with fixed end-points. 6.5. Euler’s rule. 6.6. A related problem. 6.7. The hanging string. 6.8. The chord-and-arc problem. 6.9. The classical isoperimetrical problem, Euler’s rule. 6.10. A simple eigenvalue problem. 6.11. Example of an isoperimetrical problem with a discontinuous solution. 6.12. The isoperimetrical problem with variable end-points. 6.13. Applications. 6.14. The Sturm-Liouville problem.

CHAPTER VII. Curves in space

7.1. Necessary conditions for a minimizing curve. 7.2. The corner conditions. 7.3. Existence and continuity of the second derivatives φ''(x), ψ''(x). 7.4. The extremals. 7.5. Variable end-points. 7.6. The problem of the minimizing curve with variable end-points. 7.7. Weierstrass’s necessary condition and Legendre’s necessary condition. 7.8. Fields of extremals. 7.9. Another approach to the theory of fields in space. 7.10. Jacobi’s necessary condition. 7.11. The fundamental sufficiency theorem. 7.12. Some concrete illustrations. 7.13. Problems in (n + 1) dimensions. 7.14. Lagrange’s equations deduced from Hamilton’s principle.

CHAPTER VIII. The problem of Lagrange

8.1. The problem of Lagrange. 8.2. Normality. 8.3. The equations of variation. 8.4. Lemma 1. 8.5. Lemma 2. 8.6. The first variation of I. 8.7. Necessary conditions for a minimizing curve. 8.8. The Multiplier Rule. 8.9. The matrix D. 8.10. The isoperimetrical problem. 8.11. Brachistochrone in a resisting medium. 8.12. Integrand containing y''. 8.13. Geodesics. 8.14. Lagrange’s equations deduced from the principle of Least Action. 8.15. Livens’s theorem. 8.16. A note on non-holonomic systems.

CHAPTER IX. The parametric problem

9.1. Properties of the integrand. 9.2. Relation to the ordinary problem. 9.3. Further properties of the integrand. 9.4. Necessary conditions for a weak relative minimum. 9.5. The corner conditions. 9.6. Euler’s equations. 9.7. Some concrete illustrations. 9.8. Variable end-points. 9.9. Weierstrass’s necessary condition. 9.10. Further necessary conditions. 9.11. Examples of the calculation of E. 9.12. Fields and the invariant integral. 9.13. The fundamental sufficiency theorem. 9.14. Newton’s problem. 9.15. The minimizing curve. 9.16. Proof that the solution gives a minimum value to I. 9.17. Comparison of the resistances for various curves. 9.18. The isoperimetrical problem in parametric form.

CHAPTER X. Multiple integrals

10.1. The analogue of Euler’s differential equation. 10.2. Direct methods. 10.3. Dirichlet’s principle, elementary theory. 10.4. Dirichlet’s principle for functions of two variables, introductory remarks. 10.5. Harmonic functions. 10.6. Proof of Dirichlet’s principle for the circle. 10.7. Dirichlet’s principle in the plane. 10.8. The triangle inequality. 10.9. The smoothing process. 10.10. The limit function. 10.11. Proof of Dirichlet’s principle. 10.12. Plateau’s problem. 10.13. The vibrating string. 10.14. The eigenvalues. 10.15. The vibrating membrane.

EXAMPLES

BIBLIOGRAPHY

INDEX