Wave map energy
= Wave map energy
{title2=$E=\tfrac12\int(|\phi_t|^2+|\nabla\phi|^2)\,dx$}
The conserved <energy> of a <wave map> from Minkowski spacetime to a Riemannian target is $E=\frac12\int(|\phi_t|^2+|\nabla\phi|^2)\,dx$, with norms measured in the target metric. Contraction of the <wave map> equation with the velocity and <integration by parts> gives conservation.