Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers

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


The main goal of this chapter is the derivation of sufficient conditions for one-dimensional variational problems. That is, we want to establish criteria ensuring that a given extremal u of a variational integral
$$F\left( u \right) = \int_a^b {F\left( {x,u\left( x \right),u'\left( x \right)} \right)dx} $$
is, in fact, a strong minimizer. This will be achieved by a method the elements of which were developed by Weierstrass. One of its basic ideas is to consider a whole bundle of extremals instead of a single one, just as one investigates in optics ray bundles instead of isolated single rays. This poses the problem to embed a given extremal in an entire pencil of extremals. Among bundles of extremals, those free of singularities are particularly important; they are called extremal fields. The curves of such a field, the field lines, cover some domain G of the configuration space simply, i.e., through every point of G there passes exactly one field line. It will turn out that a special kind of extremal fields satisfying certain integrability conditions will be particularly useful for the calculus of variations; these are the so-called Mayer fields. A particular feature of every such field is that it defines a scalar function S whose level surfaces form a one-parameter family of hypersurfaces which are transversal to all field curves. This function S is, up to an additive constant, uniquely determined by the field; we call it the eikonal (or optical distance function) of the field.


Nodal Point Conjugate Point Jacobi Operator Field Curve Transversal Surface 
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  1. 1.
    See I.M. Gelfand and S.V. Fomin [1], Chapter 6, and also F.H. Clarke and V. Zeidan [1]. A comprehensive treatment of matrix Riccati equation can also be found in the monograph of W.T. Reid [6].Google Scholar
  2. 2.
    Cf. Carathéodory [10], p. 295, and Bolza [3], pp. 363–364.Google Scholar
  3. 3.
    Clearly c(t) is not discontinuous in the present-day terminology but only nonsmooth, i.e., c(t) is discontinuous. The name derives from the old notation of the Leibnizian age where “continuous” roughly speaking meant: representable by analytic expressions.Google Scholar
  4. 4.
    Johann Bernoulli, Remarques sur qu’on a donné jusqu’ici de solutions des problèmes sur les isoperimetres, Mémoires de l’Acad. Roy. Sci. Paris 1718 (in Latin: Acta Eruditorum Lips. 1718). Cf. Opera Omnia, Bousquet, Lausanne and Geneve 1742, Vol. 2, Nr. CM, pp. 235–269 and Fig. 6 on p. 270, in particular pp. 266–269. Cf. also: Die Streitschriften von Jacob und Johann Bernoulli [1].Google Scholar
  5. 5.
    G. Darboux [1], Vol. 2, nr. 521.Google Scholar
  6. 6.
    For a detailed survey of some historical roots of field theory we refer to Bolza [1], in particular pp. 52–70.Google Scholar
  7. 7.
    See Hilbert [1], Problem 23, and [5]. Cf. also Bolza [3], p. 107, and [1], p. 62.Google Scholar
  8. 8.
    Disquisitiones generales circa superficies curvas, Section 15; Werke [1], Vol. 4, pp. 239–241.Google Scholar
  9. 9.
    Joh. Bernoulli, Opera omnia, torn. I, no. XXXVIII, p. 194.Google Scholar
  10. 10.
    Loc. cit., torn.. II, no. CIII, pp. 235–269; see particularly p. 267.Google Scholar
  11. 11.
    See Carathéodory [16], Vol. 5, p. 337.Google Scholar
  12. 12.
    “Durch die jetzige Veröffentlichung der Weierstraßschen Variationsrechnung hat also ein Märchen ausgelebt, an welches in den letzten Jahren fast niemand mehr geglaubt hat, das aber früher, besonders im Auslande, ziemlich verbreitet war: danach solltenich weiß nicht welche — geheimnisvolle Entdek-kungen von Weierstraß auf dem Gebiete der Variationsrechnung in seinen Papieren noch verborgen und dem mathematischen Publikum vorenthalten sein. Die Publikation des siebenten Bandes der Werke von Weierstraß kommt also zu spät, um einen erkennbaren Einfluß auf die Variationsrechnung noch zu veranlassen.” Google Scholar
  13. 13.
    Disquisitiones generales circa superficies curvas, Werke [1], Vol. 4, Section 15, pp. 239–241.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

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