Constrained variational characterization of bending modes
= Constrained variational characterization of bending modes
{title2=$W''''=k^4W$}
The <elastic energy> $U=A\int(W'')^2dx/2$ minimized without forcing gives $W''''=0$. With fixed $\int W^2dx$, stationarity of $U-Ak^4\int W^2dx/2$ gives $W''''=k^4W$: a bending eigenmode. A general admissible shape expands in these <normal modes>. The <natural boundary conditions for a free endpoint> are $W''(L)=W'''(L)=0$, while the clamp imposes $W(0)=W'(0)=0$.