# How to understand minimising length is equivalent to energy minimising?

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

• Are you looking for a proof of this statement or an explanation of what it means? Nov 13, 2015 at 13:35
• @Hrodelbert I want a proof Nov 13, 2015 at 13:37
• I think there was a proof in Calculus of Variations by Jost, but I have no access to that book right now Nov 13, 2015 at 13:40
• 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. Nov 13, 2015 at 13:41
• @HansEngler Which text has it ? I am reading $\langle$a complete proof of the differentiable 1=4-pinching sphere theorem$\rangle$. There is not. Nov 13, 2015 at 13:47

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.