I am self-studying Calculus of Variations and am struggling to prove results about the variation of a functional that are analogous to results in elementary analysis about differentials/derivatives.
Prove that if two differentiable functionals defined on the same normed linear space have the same differential (first variation) at every point of the space, then they differ by a constant.
(Gelfand & Fomin, "Calculus of Variations", Problem 1.13)
So far, I have reasoned as follows:
- Suppose $J_1[y]$ and $J_2[y]$ are functionals on the normed linear space $V$, and that $\delta J_1[y,h] = \delta J_2[y,h]$.
- Consider the functional $J[y] = J_2[y] - J_1[y]$. It follows directly from the definition that $\delta J[y,h] = \delta J_2[y,h] - \delta J_1[y,h]$ is the principal linear part of the variation $\Delta J$, and hence is the first variation of $J[y]$.
- By hypothesis, $\delta J[y,h] \equiv 0 \;\forall y\in V$.
Thus, I have reduced the original problem to proving that a functional whose variation vanishes everywhere is everywhere constant, but I can't figure out how to proceed. The common proof of the analogous result in standard calculus relies on the Mean Value Theorem, which I don't think will work here.
Pointers?