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# The First Variation

• Mariano Giaquinta
• Stefan Hildebrandt
Chapter
• 2.3k Downloads
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 310)

## Abstract

In this chapter we shall develop the formalism of the calculus of variations in simple situations. After a brief review of the necessary and sufficient conditions for extrema of ordinary functions on ℝ n , we investigate in Section 2 some of the basic necessary conditions that are to be satisfied by minimizers of variational integrals
$$\mathcal{F}(u) = \int_\Omega {F(x,u(x),Du(x)dx)}$$
(1)
.

## Keywords

Euler Equation Variational Problem Variational Integral Rotation Number Admissible Function
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## References

1. 1.
The reader may wonder why we operate with (math) instead of (math). Here we follow the custom of the theory of distributions. It has the advantage that one need not distinguish between “test functions” for differential equations of different orders. Note that, under our assumptions, (7) implies that δℱ(u,φ) holds for all (math).Google Scholar
2. 2.
For the computation of the surface area of the unit sphere in ℝn in terms of Euler’s Γ-function, (math), compare e.g. Courant-John [1] or Fleming [1].Google Scholar
3. 3.
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4. 4.
For the definition of the mean curvature, see the Supplement.Google Scholar
5. 5.
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6. 6.
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7. 7.
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9. 9.
See Supplement, nr. 6.Google Scholar
10. 10.
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11. 11.
The calculus of variations is for functionals what the differential calculus is for functions.19Google Scholar
12. 12.
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20. 20.
Par. Hist. 1787 (1789), p. 252; cf. Oeuvres 11, p. 278. In Par. Hist. 1782 (1785), p. 135 (Oeuvres 10, p. 302), Laplace had already given the formula with respect to polar coordinates.Google Scholar
21. 21.
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26. 26.
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27. 27.
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28. 28.
Mémoire sur la courbure des surfaces, Paris, Mémoires de Mathématique et de Physique (de savans étrangers) de l’Académie 10 (1785), 447–550 (lu 1776).Google Scholar
29. 29.
Methodus inveniendi lineas curvas [2], Chapter 5, nrs. 44 and 47 (pp. 194–198); cf. also Opera omnia, Ser. I, Vol. 24, pp. 182 and 185 (E65).Google Scholar
30. 30.
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31. 31.
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32. 32.
See Bolza [1], pp. 42–44, and Todhunter [1], p. 54, nr. 84, remark (3).Google Scholar
33. 33.
Jacobi’s Variations-Rechnung, handwritten notes of lectures held by Jacobi (1837/38). The notes were taken by Rosenhain. The original text is the following: Es haben sich in der neuesten Zeit die ausgezeichnetsten Mathematiker me Poisson und Gauss mit der Auffindung der Variation des Doppelintegrals beschäftigt, die wegen der willkürlichen Funktionen unendliche Schwierigkeiten macht. Dennoch wird man durch ganz gewöhnliche Aufgaben darauf geführt, z.B. durch das Problem: unter allen Oberflächen, die durch ein schiefes Viereck im Raum gelegt werden können, diejenige anzugeben, welche den kleinsten Flächeninhalt hat. Es ist mir nicht bekannt, daß schon irgend jemand daran gedacht hätte, die zweite Variation solcher Doppelintegrale zu untersuchen-, auch habe ich, trotz vieler Mühe, nur erkannt, daß der Gegenstand zu den allerschwierig-sten gehört.”Google Scholar
34. 34.
Methodus inveniendi (E65), Chapter 1, nr. 34, and Chapter 2, nr. 50.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2004

## Authors and Affiliations

• Mariano Giaquinta
• 1
• Stefan Hildebrandt
• 2
1. 1.Scuola Normale SuperiorePisaItaly
2. 2.Mathematisches InstitutUniversität BonnBonnGermany