Hamiltonian flows are challenging for data assimilation because solutions have often low regularity, they are spatially localized, and the evolution preserves certain quantities which one would like to discover and then preserve in the reconstruction. In this talk, I will present a data assimilation algorithm formulated in continuous time, and which can be implemented after time discretization. Two main ingredients of the algorithm are symplectic dynamical approximations, and a dynamic strategy to position sensors.