Turkish Journal of Mathematics
DOI
10.55730/1300-0098.3330
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 nonintegrable potential, and the Darboux-Halphen system in symmetric polynomials.
Keywords
$3D$-flows, vector potential, bi-Hamiltonian systems, Maurer-Cartan equations
First Page
3234
Last Page
3244
Recommended Citation
ESEN, OĞUL; GUHA, PARTHA; and GÜMRAL, HASAN
(2022)
"$3D$-flows generated by the curl of a vector potential & Maurer-Cartan equations,"
Turkish Journal of Mathematics: Vol. 46:
No.
8, Article 12.
https://doi.org/10.55730/1300-0098.3330
Available at:
https://journals.tubitak.gov.tr/math/vol46/iss8/12
