CoV25

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Let J:A\to\mathbb{R}\, be a functional on a subset A\, of a normed linear space V\,.

(a) Define precisely the first variation \delta J(y_0,h)\, of J\, at y_0\, and admissible h(x)\,.

\delta J(y_0, h) = \frac{d}{d\varepsilon} J(y_0 + \varepsilon h)\left.\right|_{\varepsilon = 0}

Alternatively,

\delta J(y_0, h) = \lim_{\varepsilon \to 0} \frac{J(y_0 + \varepsilon h) - J(y_0)}{\varepsilon}

The admissible h\, are \{h \in V : y_0 + \varepsilon h \in A\}

(b) Show that if \delta J(y_0,h)\, exists for a certain admissible h\isin V\,, then \delta J(y_0,\alpha h)\, also exists for every real number \alpha\,, and \delta J(y_0,\alpha h)=\alpha \delta J(y_0,h)\,.

The solution is here.


Main Page : Calculus of Variations

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