Calculus of Variations I pp 145-214 | Cite as

# General Variational Formulas

Chapter

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## Abstract

In the first two chapters we have investigated the rate of change of variational integrals ℱ with respect to variations of the dependent variables. In particular, we have derived the necessary conditions and
for extremizers u of ℱ, and suitable modifications of these conditions were shown to hold for constrained extremizers.

$$ \delta \mathcal{F}(u,\varphi ) = 0\quad for\;all\;\varphi \; \in \;C_c^\infty (\Omega ,{\mathbb{R}^\mathbb{N}})$$

$$ {L_F}(u) = 0$$

## Keywords

Vector Field Euler Equation Infinitesimal Generator Fundamental Lemma Weak Minimizer
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## References

- 1.In honour of Emmy Noether; cf.
*3, 3*and 3, 4.Google Scholar - 2.Note that, by definition, weak extremals are of class C
^{1}. Later we shall admit weak extremals which*are much*less regular. Then the regularity theorem does not always hold.Google Scholar - 3.It is throughout used in his celebrated lecture notes [1] Vol. 7; cf., in particular, pp. 107–108.Google Scholar
- 4.
- 5.Often, one adds the factor 1/2, cf.
*2,4*3 math. We have dropped it to simplify the following formulas.Google Scholar - 6.Cf. e.g. Courant-Hilbert [1–4].Google Scholar
- 7.To spare the reader any confusion, we have to remark that in the literature one often finds the notations δu =
*φ*or*δ*u = φ*for the “variation of u with respect to fixed arguments”, and δu = δu +*Du • δx*or δu =*δ*u + Du • δx*for the “total” variation of u. Similarly, one finds ∂ℱ =Φ’(0) instead of the notation (8). Our notation has the advantage that it integrates the previous definition of ∂ℱ:Google Scholar - 8.see 2 math of this section.Google Scholar
- 9.see 2 math of this section.Google Scholar
- 10.In his Königsberg lectures [4], pp. 198–211, elliptic coordinates in Rn are dealt with in full detail. For historical references, cf. Jacobi [3], Vol. 2, pp. 59–63.Google Scholar
- 11.For a presentation within the framework of differential geometry we refer the reader to the Supplement.Google Scholar
- 12.The convention about the sign of
*Δ*is not uniform, many authors prefer the opposite sign.Google Scholar - 13.
- 14.
*Mémoire sur la courbure des surfaces*, Mém. de Math. Phys. (sav. etrang.) de l’Acad.**10**(1785), 447–550 (lu 1776).Google Scholar - 15.(E385)
*Institutionum calculi integralis*, Vol. 3, Petersburg 1770, appendix; cf. also: A. Kneser,*Variationsrechnung*, Enzykl. math. Wiss., Vol. 2.1, IIA8, pp. 575–576.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2004