On the quest for generalized Hamiltonian descriptions of 3D-flows generated by the curl of a vector potential

Abstract

We examine Hamiltonian analysis of three-dimensional advection flow [Formula: see text] of incompressible nature [Formula: see text] assuming that the dynamics is generated by the curl of a vector potential [Formula: see text]. More concretely, we elaborate Nambu–Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity [Formula: see text]. We present an example (satisfying [Formula: see text]) which can be written as in the form of Nambu–Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying [Formula: see text]) which we cannot able to write it in the form of a Nambu–Hamiltonian or bi-Hamiltonian system while it can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations [Formula: see text].