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

Abstract

We study Hamiltonian analysis of three-dimensional advection flow $\dot{\mathbf{x}}=\mathbf{v}(\mathbf{x})$ of incompressible nature $\nabla \cdot \mathbf{v}=0$ assuming that dynamics is generated by the curl of a vector potential $\mathbf{v}=\nabla \times \mathbf{A}$. More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $\mathbf{A}\cdot \nabla \times \mathbf{A}$. We present an example (satisfying $\mathbf{A}\cdot \nabla \times \mathbf{A}\ne 0$) which can be written as in the form of Nambu-Hamiltonian and bi-Hamiltonian formulations. We present another example (satisfying $\mathbf{A}\cdot \nabla \times \mathbf{A}=0$) which we cannot able to write it in the form of a Nambu-Hamiltonian or bi-Hamiltonian system. On the hand, this second example can be manifested in terms of Hamiltonian one-form and yields generalized or vector Hamiltonian equations ${\dot{x}}_i=-{\epsilon }_{ijk}\partial {\eta }_j/\partial x_k$.