A Self Propelled Vortex Dipole Model on a Catenoid

Abstract: We investigate vortex dipoles on surfaces of variable negative curvature, focusing on a catenoid of arbitrary throat radius as a concrete example. We construct the effective dynamical system including mutual and geometric self-interaction terms and show that the resulting Hamiltonian dynamics makes dipoles follow catenoid geodesics, in agreement with recent works, Gustafsson (J. Nonlinear Sci. 32, 62, 2022) and by Drivas, Glukhovskiy and Khesin (Int. Math. Res. Not. 2024, 14, 10880-10894). We utilize the symplectic structure to find a conserved momentum map J related to the U(1) symmetry along the azimuthal direction. We verify the conservation of both the Hamiltonian and this momentum for arbitrary throat radius. We then demonstrate direct and exchange scattering of classical vortices on the catenoid, and we contrast this with the collective rotational motion (with azimuthal drift) that arises for chiral pairs. Finally, we build a finite-dipole dynamical system on the catenoid and show that the self-propulsion terms emerge to leading order in the dipole size. This provides a concrete realization, on a curved minimal surface, of the intuitive statement that a finite dipole propels orthogonal to the dipole axis, with a speed modulated by curvature.