Symplectic reduction of an isotropic oscillator
= Symplectic reduction of an isotropic oscillator
{title2=$H^{-1}(E)/S^1\cong\mathbb{CP}^{n-1}$}
For $H=\sum_j(q_j^2+p_j^2)/2$ and $E>0$, the oscillator flow is a free circle action on the sphere of radius $\sqrt{2E}$. Its quotient is <Complex projective space>. For two modes, the reduced form in $w=z_2/z_1$ is $iE\,dw\wedge d\bar w/(1+|w|^2)^2$, of total area $2\pi E$. Reduction at zero energy is singular instead of a smooth quotient of the asserted dimension.