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While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other subjects: for instance, one can argue that a vector space is a module over a field instead of making a "new" definition for a vector space (this is not so good of an example due to one being often introduced to vector spaces before modules, but it gets my core idea). However, when such phenomena happens in these other cases, there usually is a nice reference in the literature which makes the correspondence of definitions clear. But in all references on calculus of variations I've seen, a "variation" is a new object that is defined, and I can't see why one should not regard this as simply a case of Fréchet-derivation.

This happens even if one take a path-space of paths connecting a point $a$ to $b$, for instance. Let's consider the space $C^1([0,1], \mathbb{R}^n, a,b)$ the space of $C^1$ paths with initial point $a$ and endpoint $b$. This is an affine space over the normed vector space $C^1([0,1], \mathbb{R}^n, 0,0)$, so we have a bona-fide Fréchet-derivative, and hence we can talk about critical points. The "variations" are simply elements of the vector space.

Therefore, my questions are: Am I missing something? More precisely, is my point of view lacking or incorrect in some aspect?

If not, why isn't this approached in this way in the literature?


Sidenote: This question has been asked in MSE with an active bounty, but received an answer which I did not find sufficient for some reasons... but more objectively for the fact that it did not justify the inexistence of this point of view in the literature (or at least in the books I've seen. I'm happy to be provided a reference which does this. Even so, it seems to be rare, if it exists). I would appreciate an answer that could also address this point.

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  • $\begingroup$ Related: math.stackexchange.com/questions/1573137/… $\endgroup$ – Mikhail Katz Dec 23 '15 at 17:34
  • $\begingroup$ @katz That is my question. $\endgroup$ – Aloizio Macedo Dec 23 '15 at 17:34
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    $\begingroup$ Here are some random speculative thoughts: 1) The term "variation" is a classical one that probably came into use before the systematic use of functionals on function spaces. 2) So the computation of the Euler-Lagrange equations was originally a formal one, without any reference to a topology on the space of functions. We often still want to do this, because the right choice of what topology to use is not always obvious at the start. $\endgroup$ – Deane Yang Dec 23 '15 at 18:40
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    $\begingroup$ @Aloizio Macedo: Example: variation of the boundary of a region. Neither the boundary nor its variation is an element of a Banach space, or a vector space. $\endgroup$ – Alexandre Eremenko Dec 24 '15 at 3:12
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    $\begingroup$ But isn't the space of submanifolds with specified dimension and regularity a Banach manifold? $\endgroup$ – Deane Yang Dec 24 '15 at 6:33
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Your question actually is quite well answered around page 10 of Giaquinta and Hildebrandt's Calculus of Variations I: the Lagrangian formalism. The upshot is that the correct phrase you are looking for is the (possibly nonlinear) Gateaux differential, and not the Frechet derivative, and that is for good reason (with my apologies to French people everywhere for not being able to add accents on this device).

To quote the authors above:

Before we turn to such more concrete situations, we have to add a few remarks why we have introduced the first variation $\delta \mathscr{F}$ as fundamental notion of derivative. First, it is the oldest ... introduced by Lagrange and Euler... Secondly, and this is much more important, we prefer this notion as the fundamental one because it is so "weak". It can, for instance, happen that the first variation $\delta\mathscr{F}(u_0,\zeta)$ exists for "variations" $\zeta$ contained in some class $Z$ which is smaller than $X$ but still "sufficiently large" so that one can draw valuable conclusions on $u_0$ whereas other kinds of derivatives may not exist. Moreover, we need no norm on $X$ in order to define $\delta\mathscr{F}$ and $\delta^2\mathscr{F}$.

The authors continue with some definitions and examples to illustrate their point.

(Yes, I am aware that the Frechet derivative can be generalized to TVS, but that does not affect the bulk of the quoted explanation above.)

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