We examine the
$Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$
where
$H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$
with some regular and symmetric, but in general not explicitely known function
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Figure 1. This diagram illustrates the superposition idea. The $2$-vortex problem in the unit disc admits a stationary solution with $\Gamma^1 = -\Gamma^2$, cf. Example 1.2, say $\Gamma^1 = -2$ (blue star), $\Gamma^2 = 2$ (red star). As rigidly rotating configurations on $\mathbb{R}^2$ we take here for simplicity two identical vortices for $\Gamma^1$ and $\Gamma^2$, i.e. $\Gamma^1_1 = \Gamma^1_2 = -1$ rotate on the blue circle in clockwise direction and $\Gamma^2_1 = \Gamma^2_2 = 1$ rotate on the red circle in counterclockwise direction. The result on the right-hand side is a periodic solution of the $4$-vortex system in the disc with vorticities $\Gamma^1_1,\Gamma^1_2,\Gamma^2_1,\Gamma^2_2$, where each pair of vortices moves along a deformed circle in the same orientation as before. The shown trajectory is the actual numerically computed trajectory of the $4$-vortex problem. Suitable initial conditions can in this case be found due to symmetry considerations
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This diagram illustrates the superposition idea. The