Where Does the Hamiltonian come From?

It’s nothing but a transformation of Euler-Lagrange equation under # Legendre transformation and then Noether’s Theorem time translation symmetry.

Discrete Point Vortex Implementation

For our case of point vortex of closed surfaces. Hamiltonian is the kinetic energy of the system, which is $\int \frac{1}{2} \lvert u \rvert^2$ but in our case of $u$ is split into two parts: (1) coexact and harmonic, which splits the velocity into 2 parts as well: $H_{\text{coexact}}$ and $H_{\text{harmonic}}$.

$$H_{\text{coexact}} = \int \frac{1}{2} \psi \omega dx \implies \sum \frac{1}{2} \psi(x_{i})$$