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Weierstrass Field Theory for One-Dimensional Integrals and Strong Minimizers

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

Abstract

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.

Keywords

Nodal Point Conjugate Point Jacobi Operator Field Curve Transversal Surface 
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References

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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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