General Variational Formulas

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


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
$$ \delta \mathcal{F}(u,\varphi ) = 0\quad for\;all\;\varphi \; \in \;C_c^\infty (\Omega ,{\mathbb{R}^\mathbb{N}})$$
$$ {L_F}(u) = 0$$
for extremizers u of ℱ, and suitable modifications of these conditions were shown to hold for constrained extremizers.


Vector Field Euler Equation Infinitesimal Generator Fundamental Lemma Weak Minimizer 
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  1. 1.
    In honour of Emmy Noether; cf. 3, 3 and 3, 4.Google Scholar
  2. 2.
    Note that, by definition, weak extremals are of class C1. Later we shall admit weak extremals which are much less regular. Then the regularity theorem does not always hold.Google Scholar
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    It is throughout used in his celebrated lecture notes [1] Vol. 7; cf., in particular, pp. 107–108.Google Scholar
  4. 4.
    See the Supplement for the definition of K g . Compare also Bolza [3], p. 210.Google Scholar
  5. 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. 6.
    Cf. e.g. Courant-Hilbert [1–4].Google Scholar
  7. 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
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    see 2 math of this section.Google Scholar
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  11. 11.
    For a presentation within the framework of differential geometry we refer the reader to the Supplement.Google Scholar
  12. 12.
    The convention about the sign of Δ is not uniform, many authors prefer the opposite sign.Google Scholar
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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

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