General variations of a functional with a second order derivative?

Question

I have seen how to derive the general variation of a functional of first order. However, when I try to apply the methods to higher order functional, things break down. How does one derive the boundary conditions/ equations of motion for the general variation (I.e. endpoints also vary) of these higher order functionals?

Answer 1

The functional $f(x,y,y',y'')$ can be represented as

$$ f(x,y,y',y'')\equiv f(x,y,z,z')+\lambda(x)(z-y') = F(x,y,z,z') $$

now applying the stationary conditions

$$ F_y-\left(F_{y'}\right)' = f_y-\lambda' = 0\\ F_z-\left(F_{z'}\right)' = f_{z}+\lambda-\left(f_{z'}\right)'=0 $$

eliminating $\lambda$ we obtain

$$ f_y -\left(f_{y'}\right)'+\left(f_{y''}\right)''=0 $$

which are the Euler-Lagrange stationary conditions for the functional $f(x,y,y',y'')$