Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows

Abstract It is well known that the equations of motion of a fluid particle in a two-dimensional flow can be written in the canonical form with the stream function playing the role of the Hamiltonian. Here we extend this canonical formalism to three-dimensional steady flows. We show how the methods of the theory of Hamiltonian systems can be successfully used to investigate the advection of a passive particle. As an example of the perturbed closed streamline flow we consider a rigid rotation with added small quadratic velocity field and explain the structure of streamlines by averaging the corresponding Hamiltonian. We also show how the Hamiltonian formulation can be used to find the invariants of the fluid particle motion which are then used for non-canonical averaging.