= Angular integral in the rational map approximation
{title2=$\mathcal I[R]=(4\pi)^{-1}\int_{S^2}J_R^2d\Omega$}
For a degree-$N$ <rational map>, the <angular Jacobian of a rational map> has average $N$, so the <Cauchy-Schwarz inequality> gives $\mathcal I\geq N^2$. In dimensionless <Skyrme model> units the radial energy within the rational-map ansatz is
$$
E=4\pi\int_0^\infty\left[r^2f'^2+2N(f'^2+1)\sin^2f+\mathcal I\frac{\sin^4f}{r^2}\right]dr.
$$
First minimize $\mathcal I$ over admissible maps, then minimize over profiles with $f(0)=\pi$, $f(\infty)=0$. This gives a restricted variational approximation and an upper bound on the unrestricted minimum in that topological sector, not an exact multi-Skyrmion solution. Houghton, Manton and Sutcliffe developed this construction in https://arxiv.org/abs/hep-th/9705151.
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