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?
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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 $f(x,y,y',y'')$
Cesareo
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I'm referring to the case where the boundary points are allowed to vary as well; e.g. in the case of a functional depending only on the first derivative, you end up with a boundary term composed of the F_y'i's times the variational in y and the hammiltonian term of the system times the variational in x – Richard Wilde Jul 24 '18 at 05:55
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@RichardWilde I will try to enlarge the variation to those situations. Just now I only considered fixed end points. Thanks. – Cesareo Jul 24 '18 at 07:04