3D-flows generated by the curl of a vector potential & Maurer-Cartan equations

Esen O., Guha P., Gümral H.

Turkish Journal of Mathematics, vol.46, no.8, pp.3234-3244, 2022 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 46 Issue: 8
  • Publication Date: 2022
  • Doi Number: 10.55730/1300-0098.3330
  • Journal Name: Turkish Journal of Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.3234-3244
  • Keywords: 3D-flows, vector potential, bi-Hamiltonian systems, Maurer-Cartan equations, SYSTEMS
  • Kocaeli University Affiliated: No

Abstract

© TÜBİTAKWe examine 3D flows (Formula Presented) admitting vector identity Mv = ∇×A for a multiplier M and a potential field A. It is established that, for those systems, one can complete the vector field v into a basis fitting an 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.