The Calculus of Variations

  • I. N. Bronshtein
  • K. A. Semendyayev


The calculus of varialions concerns itself with the problem of determining from some previously given class of functions one or more functions which, in a given single or multiple integral, according to the type of functions, yields an extremum; i.e., assumes a maximal or minimal value. Many problems of theoretical physics, engineering, and geometry lead to such problems.


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  1. (1).
    This is only true in the case of the geometric variation problem where this system of equations is equivalent to one single Euler differential equation, namely, the Euler differential equation in Weierstrass form.Google Scholar
  2. (2).
    This is only the case when, as in the following example, either x(t) or y(t) does not appear in the base function.Google Scholar
  3. (3).
    To assert this is only permissible in the geometric variation problem.Google Scholar
  4. (4).
    The concept of a positive-homogeneous function is weaker than the general concept of homogeneous function. For a positive-homogeneous function the identity f(x, y, kx, ky) = kf(x, y, x, y) is fulfilled only for positive values of k, while for the general notion of homogeneity this identity must hold not only for positive but also for negative values of k. Thus, for example, the two functions \(\sqrt{{x}'^{2}+{y}'^{2}},\;\;\;{xy}'-{xy}'+\sqrt{{x}'^{2}+{y}'^{2}}\) are positive-homogeneous, but not homogeneous in the ordinary sense.Google Scholar

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© Verlag Harri Deutsch, Zürich 1973

Authors and Affiliations

  • I. N. Bronshtein
  • K. A. Semendyayev

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