Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example we evaluate these identities for a semiclassical system of two coupled oscillators.

中文翻译：

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Nambu 动力学及其在多自由度系统中的非规范哈密顿量表示

Nambu 动力学是多于两个变量的广义哈密顿动力学，其时间演化由 Nambu 括号给出，Nambu 括号是典型泊松括号的推广。通过使用 Nambu 括号定义非规范泊松括号，Nambu 动力学始终可以以非规范哈密顿动力学的形式表示。为了使时间演化保持一致，Nambu 括号必须满足基本恒等式，而非规范 Poisson 括号必须满足 Jacobi 恒等式。然而，在许多自由度系统中，众所周知，基本恒等式不成立。在本文中，我们表明，即使基本恒等式被违反，相应的非规范哈密顿动力学的雅可比恒等式也可以成立。