Hamiltonian Active Particles in Incompressible Fluid Membranes

Active proteins and membrane-bound motors exert force dipole flows along fluid interfaces and lipid bilayers. We develop a Hamiltonian framework for the interactions of pusher and puller dipoles embedded in an incompressible two-dimensional membrane supported by a shallow viscous subphase. Beginning from the Brinkman-regularized Stokes equations of the membrane-subphase system, we construct the near- and far-field dipolar velocity and associated stream functions. For two quenched dipoles, we obtain exact analytic solutions in both the near and far field regimes. Although generic dipoles reorient under the local membrane vorticity, we show that the far-field dipolar flow is vorticity-free; force-free motors therefore retain fixed orientation and obey a position-based Hamiltonian dynamics in which the positions of N dipoles evolve via an effective Hamiltonian built from the dipolar stream function. In the near field, where the flow possesses finite vorticity, a Hamiltonian formulation is recovered in the quenched-orientation limit. For identical dipoles, the far-field Hamiltonian produces rapid clustering from random initial conditions, whereas the near-field Hamiltonian suppresses collapse and yields extended, non-aggregating configurations. Our work thus provides a concrete realization of position-based Hamiltonian descriptions for active particles in incompressible fluid membranes and shows that hydrodynamic screening alters not only the interaction range but also the phase-space structure, integrable dynamics, and collective organization of active dipoles.