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.

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 and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.