Question

Derive the Euler-Lagrange equations and the associated boundary conditions the functional (Reference homework solution to give details of the processing) 1.

Answer 1

Given functional is π(y) (ry(x))dr

5% + tz)"2(x) + q(y + tz)2(x)] dr tz) = For given t E R, and a function define f(t) = π(y+

Let us denote the numerator and the denominator as g(t), h(t) respectively

To compute the Euler-Lagrange equation we consider 0 0 since t 0 is the critical point of f when y is a critical point of π

0 ) f'(0) = g()-g(0)h'(0 f(0)0

$g'(0)=\frac{d}{dt}\int_0^L\left[S(y+tz)''^2(x) +q(y+tz)^2(x)\right ]dx\Big|_{t=0}$

In に0

Under Dirichlet boundary conditions that: y"(0)-0-y"(L): y(0) y(L), and using integration by parts, we get in

t=0 t=0

Using integration by parts, we get

$g(0)=\int_0^L(Sy''^2+qy^2)\,dx \text{ and }h(0)=\int_0^Lry'\,dx.$.

$\text{Now }f'(0)=0 \text{ implies that }\\$

$2\int_0^L (Sy^{(4)}+qy)z\,dx +2\frac{g(0)}{h(0)}\int_0^L ry''z\,dx=0 \\ \Rightarrow \int_0^L (Sy^{(4)}+\lambda ry''+qy)z\,dx=0,\ \left(\text{where }\lambda=\frac{g(0)}{h(0)}\right).$

$\text{Therefore, the Euler-Lagrange equation is }$

$Sy^{(4)}+\lambda ry''+qy=0.$