Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier (Q326674)

This short and self-contained paper deals with a system of first order ordinary differential equations which is completely integrable. The system can be put into the Nambu formalism (which is extensively explained in the appendix) via a coordinate transformation where the transformation is shown to be a Jacobian multiplier if the system is not divergence-free. Furthermore, it is stated that there are as many independent Jacobi multipliers as independent first integrals of the system. This goes along with previous statements proposed in [\textit{J. Llibre} et al., J. Nonlinear Sci. 25, No. 4, 815--826 (2015; Zbl 1332.34004)]. In Section 3, some very nice applications of the formalism are provided. The generalization of Hamiltanian dynamics used throughout this paper as introduced by Nambu in 1973 (and therefore called ``Nambu dynamics'') is elaborated in the appendix and the relevant concepts of Nambu-Hamiltonian dynamics -- like the ``Nambu bracket'' as generalization of the Poisson bracket -- are explained.