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 fields and associated stream functions. For two quenched (fixed orientation) dipoles, we obtain exact analytic solutions in both the near- and far-field regimes. Although generic dipoles reorient under local membrane vorticity, we show that the far-field dipolar flow is vorticity-free; force-free motors therefore retain fixed orientations 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 leading to nonaggregating configurations. Our work thus provides a concrete realization of a position-based Hamiltonian formulation for active particles in incompressible fluid membranes and shows that hydrodynamic screening can reorganize both the dynamical phase space and the collective organization of active dipoles.