# Calculus of Variations I. The Langrangian Formalism: The by Mariano Giaquinta, Stefan Hildebrandt

By Mariano Giaquinta, Stefan Hildebrandt

This long-awaited publication through of the most important researchers and writers within the box is the 1st a part of a treatise that would disguise the topic in breadth and intensity, paying certain recognition to the ancient origins, partially in purposes, e.g. from geometrical optics, of elements of the speculation. numerous aids to the reader are supplied: the certain desk of contents, an creation to every bankruptcy, part and subsection, an summary of the appropriate literature (in Vol. 2) plus the references within the Scholia to every bankruptcy, within the (historical) footnotes, and within the bibliography, and eventually an index of the examples used during the booklet. Later volumes will take care of direct equipment and regularity conception. either separately and jointly those volumes will surely develop into average references.

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**Example text**

9t(u) = u(x), Du(x)) dx, fa which will be called variational integrals. 0ro(u) or F(u, £1) if we want to indicate the domain of integration 0. em(u) will be denoted as Lagrangian, or variational integrand, or Lagrange function. The reader will have noticed that we denote the variational integrals that are associated to Lagrangians F, G, ... by the script types A,1, ... of the same letters. We will use this convention whenever it does not lead to some misinterpretation. Since one has to distinguish between the Lagrangian F(x, u, p) as function of the independent variables x, u, p, and the composed function F(x, u(x), Du(x)), it would be more precise to write F(x, z, p) instead of F(x, u, p).

Proof. For f, g e L2(Q), we have the following geometric interpretation of the assertion: Suppose that f is orthogonal in L2(Q) to all functions W which are perpendicular to g and of class q(Q). , f = Ag for some A e R. This interpretation suggests also the proof that even works in the general case. If (ss) holds for all tP e Cm(Q), then the assertion follows at once from Lemma 3. Hence we can assume that there is some Vi a C'(0) such that g(x)o(x) dx # 0 So and, multiplying 0 by some suitable constant, we have that there is some 0 e CC°(Q) with

W) = J e a o A variation y(x, e) of u(x) which is not subject to any boundary condition on BR will be called a e 0 X Fig. 6. (x, e) allow the variations to satisfy nonlinear constraints. 14 Chapter 1. The First Variation variation with free boundary values. If, however, the additional condition (iii) O(x, e) = u(x) for x e aQ and IaI < ao is satisfied, we speak of a variation with fixed boundary values. Its velocity vector ip(x) - ik,(x, 0) fulfils p(x) = 0 for x e M. Variations of u(x) that are of the more general type #(x, a) have to be used if we consider variational problems where the comparison functions are subjected to nonlinear constraints.