Symbolic test of the Jacobi identity for given generalized 'Poisson' bracket

We have developed and provide an algorithm which allows to test the Jacobi identity for a given generalized 'Poisson' bracket. Novel frameworks for no nequilibrium thermodynamics have been established, which require that the r eversible part of motion of thermodynamically admissible models is describe d by Poisson brackets satisfying the Jacobi identity in order to ensure the full time-structure invariance of equations of motion for arbitrary functi on(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, Bur the manual evaluation of Jacobi identities relevant for applications and ev en for basic examples is often very time consuming. The efficient algorithm presented here can be obtained as a package to be used within the framewor k of the symbolic programming language Mathematical (TM) The tool handles P oisson brackets acting either on functions or on functionals, depending on whether the system is described in terms of discrete or of continuous varia bles, (C) 2001 Elsevier Science B.V, All rights reserved.