= Moment map for rotation of the complex projective line
{title2=$H(z)=|z|^2/(1+|z|^2)$}
For the <symplectic form> $\omega=i\,dz\wedge d\bar z/(1+|z|^2)^2$, the <circle group> rotates $z$ by $e^{i\theta}$. Its real generator is $K=iz\partial_z-i\bar z\partial_{\bar z}$. The contraction is $\iota_K\omega=-dH$, so $H$ is a <Hamiltonian function> in this sign convention. In the other chart $w=1/z$, $H=1/(1+|w|^2)$, proving smoothness at infinity. Its image is $[0,1]$ with the endpoints at the fixed poles. The <symplectic area> is $2\pi$; rescaling to the round area $4\pi$ would double this moment function.
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