Example II

Find Euler$-$Lagrange equations for

$I(v)={\displaystyle \underset{0}{\overset{L}{}}}[\frac{1}{2}A{v}^2,+\frac{1}{2}K{v}^2-q(x)v]dx-Mv{v}_x(L)$

$v(0)=v(L)=0$

$(A{v}_{,}{)}_{}+Kv=q$

$[Av,{?}_{}{]}_x=0=0$

$[A{v}_{}{]}_x=L=M$

$v(0)=v(L)=0$

What physical problem can this represent?

D Compute the weak from starting from the Euler$-$Lagrange equations.