$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

TL;DR

This paper explores 3D flows with vector potentials, revealing their algebraic structure, potential Hamiltonian properties, and providing concrete examples to illustrate the theoretical framework.

Contribution

It introduces a novel algebraic and geometric framework for analyzing 3D flows with vector potentials, including their Maurer-Cartan structure and Hamiltonian characteristics.

Findings

Flows can be completed into an $rak{sl}(2)$ algebraic basis.

The structure equations are expressed in Maurer-Cartan form.

Some systems exhibit bi-Hamiltonian structure with first integrals.

Abstract

We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vector identity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and a potential field $\mathbf{A}$. It is established that, for those systems, one can complete the vector field $\mathbf{v}$ into a basis fitting an $\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some concrete cases, three examples are provided, namely the Gulliot system, a system with a non-integrable potential, and the Darboux-Halphen system in symmetric polynomials.