I'm trying to derive the Euler-Lagrange equation from the Wilmore functional $\displaystyle W(\Sigma) = \int_{\Sigma} H^2\cdot dA$, where $\Sigma$ is some surface. The resulting equation should be
$$\Delta H + 2H(H^2-K) = 0,$$
here $H$ denotes the mean curvature, $K$ is the Gaussian curvature and $\Delta$ denotes the Laplace–Beltrami operator. I'm really clueless on how one would obtain a derivation of this in $\mathbb{R}^3$ with some equipped metric. There's a proof on page 2 here but it uses a lot of heavy machinery which is difficult for me to follow. Is it possible to derive the said expression using the basic principles from calculus of variations? I've only taken an introductory course on differential geometry and self studied basic calculus of variations.