Abstract
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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Notes
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.
This is only the case when, as in the following example, either x(t) or y(t) does not appear in the base function.
To assert this is only permissible in the geometric variation problem.
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.
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© 1973 Verlag Harri Deutsch, Zürich
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Bronshtein, I.N., Semendyayev, K.A. (1973). The Calculus of Variations. In: A Guide Book to Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-6288-3_15
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DOI: https://doi.org/10.1007/978-1-4684-6288-3_15
Publisher Name: Springer, New York, NY
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