I have a question regarding the below reasoning. The only relevant information that is missing from this picture is that $\mathbf{\Omega}(\tau) = \sin \theta \cos \phi\mathbf{i}+\sin \theta \sin \phi \mathbf{j} + \cos \theta \mathbf{k}$, or the standard unit vector on the 2-sphere. What I can't figure out for the life of me is how one reduces "$\delta S=0$" to "$\mathbf{\Omega} \times \mathbf{\dot{\Omega}} = -\mathbf{h}(\tau)$".
In particular, it's the presence of vectors in the integrand that is making it difficult for me to compute the first variation. Any help will be greatly appreciated!
[Please note that whilst very similar to a recent post of mine, the question I have here is slightly different.]
