Proof. [0233]

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Proof.

We differentiate

	$\displaystyle\frac{d}{ds}\left(\frac{(n-1)}{n}\left(\frac{d\mathfrak{w}}{ds}\right)^{n}+\frac{1}{2\mathfrak{w}^{2}}\right)$	$\displaystyle=(n-1)\left(\frac{d\mathfrak{w}}{ds}\right)^{n-1}\left(\frac{d^{2}\mathfrak{w}}{ds^{2}}\right)-\mathfrak{w}^{-3}\frac{d\mathfrak{w}}{ds}$
		$\displaystyle=\left(\frac{d\mathfrak{w}}{ds}\right)\left(\frac{d}{ds}\left(\frac{d\mathfrak{w}}{ds}\right)^{n-1}-\mathfrak{w}^{-3}\right)$
		$\displaystyle=0$

Now the result follows by observing that, from the initial conditions we have

\left(\frac{(n-1)}{n}\left(\frac{d\mathfrak{w}}{ds}\right)^{n}+\frac{1}{2\mathfrak{w}^{2}}\right)\bigg|_{s=1}=\frac{1}{2w_{0}^{2}}.

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