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

Abstract

We examine 3D flows x = v(x) admitting vector identity Mv = backward difference x 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.