Point Vortex Dynamics

Kirchhoff (1876) — Hamiltonian vortex motion, integrable N≤3, chaotic N≥4

N=3 | H = — | Click canvas or use buttons
Point vortex dynamics: N vortices with circulations κᵢ move according to ẋᵢ = (1/2π) Σⱼ≠ᵢ κⱼ (yᵢ−yⱼ)/rᵢⱼ², ẏᵢ = −(1/2π) Σⱼ≠ᵢ κⱼ (xᵢ−xⱼ)/rᵢⱼ². The system is Hamiltonian with H = −Σᵢ<ⱼ κᵢκⱼ ln|rᵢⱼ|/2π. N≤3: integrable (conserved quantities: H, linear/angular impulse). N≥4: generically chaotic. Click canvas to add vortices. Cyan/blue = positive circulation; Red/orange = negative.