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

## TL;DR

This paper investigates Hamiltonian structures of 3D incompressible flows generated by the curl of a vector potential, exploring conditions for Nambu-Hamiltonian and bi-Hamiltonian formulations and providing specific examples.

## Contribution

It analyzes the Hamiltonian nature of 3D flows based on the vector potential's properties, introducing new examples and limitations of such formulations.

## Key findings

- Example with non-zero A·∇×A admits Nambu-Hamiltonian form.
- Example with zero A·∇×A cannot be expressed as Nambu-Hamiltonian.
- Generalized Hamiltonian equations are derived for the second example.

## Abstract

We study Hamiltonian analysis of three-dimensional advection flow $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ of incompressible nature $\nabla \cdot {\bf v} ={\bf 0}$ 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 $\mathbf{A} \cdot \nabla \times \mathbf{A} \neq 0$) 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 $\dot{x}_i = - \epsilon_{ijk}{\partial \eta_j}/{\partial x_k}$.