I have a problem connected to finding a minimal surface with a given boundary. I know that it is the surface with zero mean curvature but as I have to obain differential equations for such surface I decided to do it by looking for extremal points of area functional:
$$ S= \int\limits_{surface} \sqrt{\text{det}(g_{ind})} $$ where $g_{ind}$ is the metric induced on my 2d surface from (in my case) rather nontrivial 4 dimensional metric (Einstein summation convention from now on):
$$ g_{ind, \mu \nu}=g_{\alpha, \beta}\frac{\partial X^\alpha}{\partial \sigma_\mu}\frac{\partial X^\beta}{\partial \sigma_\nu} $$ Where $\sigma^\nu, \nu\in {1,2}$ are the coordinates on surface. to the functional $S$ I applied standard Euler-Lagrange equations. But then I thought that the standard trick when looking for minimal curves: rather then minimise the length functional: $$ \int\limits_{curve} \sqrt{g_{\mu \nu}\frac{d X^\mu}{d t}\frac{d X^\nu}{d t}} $$
one can minimise
$$ \int\limits_{curve} g_{\mu \nu}\frac{d X^\mu}{d t}\frac{d X^\nu}{d t} $$ and obtain identical set of equations. Wikipedia states that minimising those two is equavalent due to Swartz inequality. So I tried the trick with my S functional and minimise such integral
$$ \tilde{S}= \int\limits_{surface} \text{det}(g_{ind}) $$
instead, but it turned out that the set of equations now obtained is not equivalent to those obtained by ninimising $S$. Can anyone explain to me why this does not work? does it mean that I have made some errors in my calculation? Or just the 2d case does not allow this simplification?