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

机译：给定广义“泊松”括号的雅可比身份的符号测试

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摘要

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 Mathematica~(TM). 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.
机译：我们已经开发并提供了一种算法，该算法可以针对给定的广义“泊松”括号测试雅可比身份。已经建立了非平衡热力学的新框架，该框架要求热力学可容许模型的运动的可逆部分由满足雅可比（Jacobi）身份的泊松括号描述，以确保任意函数的运动方程的全时结构不变性在状态空间上。对于诸如李括号之类的对象服从的非缔合代数，李群的元素满足该身份。但是，手动评估与应用程序甚至基本示例相关的Jacobi身份通常非常耗时。此处提出的有效算法可以作为在符号编程语言MathematicaTM的框架内使用的软件包而获得。该工具处理作用在功能上或作用在功能上的泊松括号，这取决于系统是用离散变量还是连续变量来描述的。

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