I have been looking for a rigorous proof of Euler's equation, in the following context. Let J be a functional of the form \begin{equation} J(y) = \int_a^b F(x,y(x),y'(x))\,\text{d}x, \end{equation} defined on all continuously differentiable functions $y:[a,b]\to \mathbb R$ that satisfy $y(a)=A$, $y(b)=B$, for fixed numbers $A,B$. Then if $J$ has an extremum at $y$, then Euler's equation must hold: \begin{equation} \frac{\partial F}{\partial y} = \frac{\text{d}}{\text{d}{x}}\left(\frac{\partial F}{\partial y'}\right). \end{equation} I found a proof of this on page 14-15 of Gelfand and Fomin's Variational calculus (which can be read online at http://users.uoa.gr/~pjioannou/mech2/READING/Gelfand_Fomin_Calculus_of_Variations.pdf), but I do not follow their logic competely.
In the last sentence before equation 12 they write, `It follows by Taylor's theorem that $\dots$', and then, in eq. 12, they write the function $F$ (as I introduced above) as its Taylor series. I don't see how this is justified, for it is not assumed that $F$ is analytic, and it might not be equal to its Taylor series.
My question is twofold.
$(1)$ Is the proof of Gelmand and Fomin valid for non-analytic functions $F$? And if it is, then how does this relate to the fact that $F$ might not equal its Taylor series?
(2) Does anyone know of different rigorous proofs of the proposition? And where can I find those?
Thanks!
$F$ is not assumed 'analytic' in their text. It suffices for their equation (12) that $F$ is $C^1$ in its arguments. They only use a local expansion. And then for the subsequent integration by part that it has continuous 2nd order derivatives. This is also stated in the beginning of 4.1. You may take a look in Euler-Lagrange which essentially use the same kind of arguments. There is nothing particularly deep in the arguments. The real difficulty comes if you want to show existence of a critical point for which you typically assume some kind of convexity in the last argument. I don't know of any easy account of that.