I am studying the dynamics of vortices in simply connected domains, in other words, in regions that are conformally equivalent to the circle.
Then, through the theory of conformal maps, the Hamiltonian that provides the dynamics in such domains is given by the theorem in the image below.
It is known in general that the Hamiltonian that describes the movement of vortices in regions with boundarie is given through the Green's function of the first type, where we can decompose it into the sum of two parcels, where the first is related to the interaction of the vortices with the boundarie and the second the interaction of the vortices among themselves.
In this material where the image was captured (N-problem Vortices, author: Paul. Newton), a simplified version of this theorem is made (the case for only one vortex), I did it and I believe I understood this case well.
In the simple case, he starts by considering a conformal map between two planes, and for the plane of the map's domain, we have what we call an associated complex potential, where the imaginary part of the potential (the potential is an analytic function) gives us the trajectory of the vortex, through the contour lines.
Well, using the conformal inverse map, it is possible, in addition to explicitly establishing the complex potential associated with the codomain plane, to see the relationship between such potentials. And that allows us to take the next step, establishing the complex velocity at the place where the vortex is (which is given through the derivative of the complex potential with respect to the considered variable).
The starting point to overcome the second step is to consider the complex potential for flows induced by a single vortex, which is given in the literature by: $$w(z)=-\dfrac{i \Gamma}{2 \pi}log(z-z_0)$$
And then, we look at the potential associated with the flow without yet considering the vortex at the point $z_0$, we take the potential associated with the flow without the point $z_0=(x_0,y_0)$, we use Taylor expansion in a neighborhood of $z_0$ for the map according to $f$, this is possible, because the map is analytic, in particular it has a derivative at such a point. This completes step two.
And in the last step, we just use step two and the fact that the Hamiltonian is the imaginary part of the complex potential (minus the singularities), and now instead of considering the whole plane, we restrict it to the region we want to study and through the Riemannian Application Theorem, it is still possible to maintain the relation obtained in the previous step, but now for a simply connected domain that is mapped on the circle (via Riemann) we have the relation between the Hamiltonians: \begin{equation} H(x_0,y_0)=\mathcal{H}(\xi, \eta )- \dfrac{ \Gamma}{4 \pi} \log\left|\left|f'(z_0)\right|\right|, \end{equation}
But the Hamiltonian for the circle case is well known in the literature via the method of images, given by \begin{equation} \mathcal{H}(\xi_0, \eta_0 )= - \dfrac{ \Gamma}{4 \pi} \log \left(\dfrac{1}{\left| \left|\zeta_0 \overline{\zeta_0}-1\right| \right|}\right), \end{equation} where $\zeta_0=\xi_0 + i \eta_0$.
Thus, the expression for the simple version (with a single vortex): $$H_{z_0}(x_0,y_0)=- \dfrac{ \Gamma}{4 \pi} \log \left(\dfrac{\left|\left|f'(z_0)\right|\right|}{1-\left|\left|f(z_0)\right|\right|^2}\right).$$
For the general case, I have a problem. Once I have more than one vortex in a closed domain, I no longer have just the interaction of the vortex with the boundary (interaction, I mean the velocity field induced by a vortex in the boundary, where the boundary "bounces" this field, inducing the velocity of the fluid particles in the domain, including the velocity of the vortex itself), I also have the interaction of the vortices among themselves, that is, a vortex induces a velocity field in the other and vice versa, influencing the trajectory of the vortices within the closed and simply connected domain, this is represented by the expression for the Hamiltonian in the image; the first summation refers to the first type of interaction already mentioned, and the double summation refers to the second type of vortex-vortex interaction.
Well then, for the general case of N-vortices, repeating the idea for the case of a vortex in each of the N-vortices and adding the contributions, we obtain exactly the first summation. But I can't come up with ideas to make the double summation appear even for the case of two vortices. If anyone can point a way, give any advice, I would greatly appreciate it.