Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.2: Ordinary Differential Equations

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.

\(\Box\)

Contributors