1.2: Ordinary Differential Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
Set
$$E(v)=\int_a^bf(x,v(x),v'(x))\ dx\]
and for given
$$V=\{v\in C^2[a,b]:\ v(a)=u_a,\ v(b)=u_b\},\]
where
(P)
Euler equation
Let
$$\frac{d}{dx}f_{u'}(x,u(x),u'(x))=f_u(x,u(x),u'(x))\]
in
Exercise
For fixed
Figure 1.2.1.1: Admissible Variations
Basic lemma in the calculus of variations. Let
for all
Proof. Assume
Set
$$
\phi(x)
=\left\{
\]
Thus
$$\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.
Contributors and Attributions
Integrated by Justin Marshall.