= Rotating point-vortex relative equilibrium
{title2=$d(H+\Omega L)=0$}
For identical <point vortices> on the plane, use $\omega=\sum_a dx_a\wedge dy_a$, $\iota_{X_H}\omega=dH$, and $L=\tfrac12\sum_a(x_a^2+y_a^2)$. The generator $Y$ of simultaneous counterclockwise rotations is $-X_L$, and $L$ is its <moment map> with the convention $\iota_Y\omega=-dL$. In coordinates rotating with <angular velocity> $\Omega$, the <Hamiltonian vector field> is $X_H-\Omega Y=X_{H+\Omega L}$. Hence a collision-free configuration rigidly rotating at that velocity is exactly a <critical point> of the augmented <Hamiltonian function> $H+\Omega L$. For $H=-\sum_{a<b}\log|\mathbf r_a-\mathbf r_b|^2$, scaling all positions shows $\sum_a\mathbf r_a\cdot\nabla_aH=-N(N-1)$; a rotating relative equilibrium therefore obeys $2\Omega L=N(N-1)$. For $N\geq2$ this implies positive $\Omega$ in the stated circulation convention. Reversing the contraction convention requires consistent changes to the Hamiltonian and rotation signs.
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