One solution of the 3D Jacobi identities allows determining an infinity of them

Abstract

It is demonstrated that the knowledge of a single and arbitrary solution of the three-dimensional Jacobi equations allows determining infinite families of new solutions, which are generally and explicitly constructed in what follows. Examples are given.

On time-dependent Hamiltonian realizations of planar and nonplanar systems

“…The general solution of the Jacobi identity (44) is given by (1/M )∇H for two arbitrary functions M and H [2,24,25,26]. Here, the existence of a Jacobi's last multiplier M is a manifestation of conformal invariance of the Jacobi identity [16,28,29,56].…”

On integrals, Hamiltonian and metriplectic formulations of polynomial systems in 3D

“…The following theorem establishes form of a general solution of the Jacobi identity. For the proof this theorem we refer [1,31,32,33].…”

Characterization, global analysis and integrability of a family of Poisson structures

“…Thus, making use of (22) in equation (16), the transformation dτ = J 12 (x(y))dt is performed. According to (12)(13)(14) this leads from the structure matrix (21) to the Darboux canonical one:…”