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How to understand the equivalent above red line in picture below ?

enter image description here

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  • $\begingroup$ Are you looking for a proof of this statement or an explanation of what it means? $\endgroup$
    – Hrodelbert
    Nov 13, 2015 at 13:35
  • $\begingroup$ @Hrodelbert I want a proof $\endgroup$
    – Enhao Lan
    Nov 13, 2015 at 13:37
  • $\begingroup$ I think there was a proof in Calculus of Variations by Jost, but I have no access to that book right now $\endgroup$
    – Hrodelbert
    Nov 13, 2015 at 13:40
  • $\begingroup$ The argument given in the text is clearly not sufficient to show that the two minimization problems are equivalent. For example $\sin x \le 1 + x^2$ for all $x$, but the minima are clearly different. What needs to be shown is that the Euler Lagrange equations for the two problems have the same solution sets and that minimizers are unique. $\endgroup$ Nov 13, 2015 at 13:41
  • $\begingroup$ @HansEngler Which text has it ? I am reading $\langle$a complete proof of the differentiable 1=4-pinching sphere theorem$\rangle$. There is not. $\endgroup$
    – Enhao Lan
    Nov 13, 2015 at 13:47

1 Answer 1

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Here is an idea of a proof:

For any curve we can choose a reparametrisation such that its speed is constant, such that we actually have $L(\gamma)^2 = 2 E(\gamma)$ (this follows from going through the Cauchy-Schwarz inequality that already underlies the inequality in the text) and we only have to consider curves with constant speed. Now it is clear that if any of those curves minimizes $E$ it also minimizes $L$ and since $L$ is always positive the converse is also clear.

So I would say that the text is somewhat misleading in saying that they have provided all the necessary ingredients to prove this statement, but it is true.

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  • $\begingroup$ Thanks ,let parameter be arc length ,then I finish it according your answer. $\endgroup$
    – Enhao Lan
    Nov 13, 2015 at 13:52

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