Find a function $\phi$ of class $C^2$ (first and second derivatives exist and are continuous) that minimice the functional:
$I(\phi)= \int_0^1 \frac {\phi''(t)} {\phi(t)} dt$
and $\phi(0)=1$, $\phi(1)=4$.
I think i have to use Euler lagrange equations to solve this. But how? The functional has second derivative of $\phi$! Any help would be greatly appreciated!
Best Answer
Hint: Euler-Lagrange equations for a Lagrangian $L(\phi,\dot{\phi},\ddot{\phi},\dots)$ are given by $$\frac{\partial L}{\partial\phi}-\frac{d}{dt}\Big(\frac{\partial L}{\partial\dot{\phi}}\Big)+\cdots+(-1)^n\frac{d^n}{dt^n}\Big(\frac{\partial L}{\partial\phi^{(n)}}\Big)=0.$$