Symbolic test of the Jacobi identity for given generalized ‘Poisson’ bracket Original Research Article

We have developed and provide an algorithm which allows to test the Jacobi identity for a given generalized ‘Poisson’ bracket. Novel frameworks for nonequilibrium thermodynamics have been established, which require that the reversible part of motion of thermodynamically admissible models is described by Poisson brackets satisfying the Jacobi identity in order to ensure the full time-structure invariance of equations of motion for arbitrary function(al)s on state space. For a nonassociative algebra obeyed by objects such as the Lie bracket, the elements of Lie groups fulfill this identity. But the manual evaluation of Jacobi identities relevant for applications and even for basic examples is often very time consuming. The efficient algorithm presented here can be obtained as a package to be used within the framework of the symbolic programming language MathematicaTM. The tool handles Poisson brackets acting either on functions or on functionals, depending on whether the system is described in terms of discrete or of continuous variables.