Example $4\mathrm{.}5-4.$ Find the equations that must $be$ satisfied $byan$ extremal

for the functional

$J(w)={\displaystyle \underset{t_0}{\overset{ts}{}}}\frac{1}{2}[w_1^2(t)+w_2^2(t)]dt,$

where the functions $w_1$ and $w_2$ are related $by$

$w_1^{}(t)=w_2(t).$

There $is$ one constraint, $so$ the function $finEq.(4\mathrm{.}5-41)is$

$f(w(t),w^{}(t))=w_2(t)-w_1^{}(t),$

and one Lagrange multiplier $p(t)is$ required. The function $g_{a}inEq.$

$(4\mathrm{.}5-43)is$

$g_a(w(t),w^{}(t),p(t))=\frac{1}{2}{w}_1^2(t)+\frac{1}{2}{w}_2^2(t)+p(t)w_2(t)-p(t)w_1^{}(t)$

From $Eq.(4\mathrm{.}5-42a)we$ have

$w_1^{*}(t)+p^{}{?}^{*}(t)=0$

$w_2^{*}(t)+p^{*}(t)=0,$

and satisfaction $of(4\mathrm{.}5-46)$ requires that

$w_1^{}{?}^{*}(t)=w_2^{*}(t).$