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Mathematics LibreTexts

1.2: Ordinary Differential Equations

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  • Page ID
    2130
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    Set

    $$E(v)=\int_a^bf(x,v(x),v'(x))\ dx$$

    and for given \(u_a,\ u_b\in{\mathbb R}\)

    $$V=\{v\in C^2[a,b]:\ v(a)=u_a,\ v(b)=u_b\},$$

    where \(y\) and \(f\) is sufficiently regular. One of the basic problems in the calculus of variation is


    (P)                \(\min_{v\in V}E(v)\).

    Euler equation

    Let \(u\in V\) be a solution of (P), then

    $$\frac{d}{dx}f_{u'}(x,u(x),u'(x))=f_u(x,u(x),u'(x))$$

    in \((a,b)\).

    Exercise \(\PageIndex{1}\): Proof

    For fixed \(\phi\in C^2[a,b]\) with \(\phi(a)=\phi(b)=0\) and real \(\epsilon\), \(|\epsilon|<\epsilon_0\), set \(g(\epsilon)=E(u+\epsilon \phi)\). Since \(g(0)\le g(\epsilon)\) it follows \(g'(0)=0\). Integration by parts in the formula for \(g'(0)\) and the following basic lemma in the calculus of variations imply Euler's equation.

    Admissible Variations

    Figure 1.2.1.1: Admissible Variations

    Basic lemma in the calculus of variations. Let \(h\in C(a,b)\) and

    $$\int_a^bh(x)\phi(x)\ dx=0$$

    for all \(\phi\in C_0^1(a,b)\). Then \(h(x)\equiv0\) on \((a,b)\).

    Proof. Assume \(h(x_0)>0\) for an \(x_0\in (a,b)\), then there is a \(\delta>0\) such that \((x_0-\delta,x_0+\delta)\subset(a,b)\) and \(h(x)\ge h(x_0)/2\) on \((x_0-\delta,x_0+\delta)\).
    Set

    $$
    \phi(x)
    =\left\{\begin{array}{r@{\quad\mbox{if}\quad}l}
      \left(\delta^2-|x-x_0|^2\right)^2 & x\in(x_0-\delta,x_0+\delta)\\
      0   & x\in (a,b)\setminus[x_0-\delta,x_0+\delta]
      \end{array} \right. .
    $$

    Thus \(\phi\in C_0^1(a,b)\) and

    $$\int_a^b h(x)\phi(x)\ dx\ge \frac{h(x_0)}{2}\int_{x_0-\delta}^{x_0+\delta}\phi(x)\ dx>0,$$

    which is a contradiction to the assumption of the lemma.

    \(\Box\)

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