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The lemma of du Bois-Reymond as given in the textbook "A fist course in variational calculus" by Mark Kot:

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The proof is as follows: enter image description here

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The proof uses a specific variation $\eta(x)$, but I'm struggling to see how this generalizes to all possible legal variations. Can someone explain?

Thanks a lot.

Bernard
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Dahlai
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  • What exactly is your question? I don't understand what you want to generalize. The statement is about $M$ being constant under some assumptions. In the proof we see that the assumptions could actually weakend to require the equality $(2.63)$ only for one particular variation (i.e. we couldn't care less what happens with general variations, because we don't need them to prove the statement we want). – Severin Schraven Aug 10 '19 at 10:31
  • @SeverinSchraven That's exactly the point I'm struggling with. Where exactly do we weaken our assumptions? In my mind the proof shows that if we take some particular variation, $M(x)$ comes out as having to be some constant, but I don't see how this generalizes beyond the chosen variation. – Dahlai Aug 11 '19 at 15:16
  • It is not needed to generalize anything. We assume $(2.63)$ to hold true for all variations. In particular it is true for the particular one we have chosen in the proof. That is enough to extract the information that $M$ is constant. What else do you want to show? – Severin Schraven Aug 11 '19 at 16:28

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