On foundation of the generalized Nambu mechanics
On foundation of the generalized Nambu mechanics
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DOI：
10.1007/bf02103278
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发表时间：
1993-01
影响因子：
2.4
通讯作者：
L. Takhtajan
中科院分区：
文献类型：
--
作者：
L. Takhtajan
我们概述了一个典型的形式主义的Nambu力学的基本原则，一个推广的哈密顿力学提出的Yoichiro Nambu在1973年。它是基于一个Nambu括号的概念，它概括了泊松括号-一个“二进制”操作的经典可观的相空间的“多”操作的高阶n × 3。Nambu动力学由Nambu-Hamilton运动方程给出的相流描述，这是一个包含n −1个“Hamilton”的常微分方程系统。我们介绍的基本身份的Nambu括号的雅可比身份的推广作为一致性条件的动态。我们表明，Nambu括号结构定义了一个层次的无限家庭的“从属”结构的低阶，包括泊松括号结构，满足一定的匹配条件。Nambu括号的概念使我们能够为Nambu力学定义Nambu-Poisson流形-相空间，这比Hamilton力学的Poisson流形-相空间更“刚性”。介绍了南武力学的作用形式和作用原理的类比。在它的表述中，循环的动力学（在一般的n元情况下是n−2维链）自然出现。我们讨论了几种方法的量子化的南部力学，变形理论的基础上，路径积分制定和南部海森堡“对易”关系。在后一种形式中，我们给出了Nambu-Heisenberg关系在then=3情况下的一个显式表示。我们强调了三元和高阶代数运算以及与它们相关的数学结构在从汉密尔顿的动态图像到南部的动态图像的传递中所发挥的作用。
We outline basic principles of a canonical formalism for the Nambu mechanics—a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of a Nambu bracket, which generalizes the Poisson bracket—a “binary” operation on classical observables on the phase space—to the “multiple” operation of higher ordern≧3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion—a system of ODE's which involvesn−1 “Hamiltonians.” We introduce the fundamental identity for the Nambu bracket—a generalization of the Jacobi identity—as a consistency condition for the dynamics. We show that Nambu bracket structure defines a hierarchy of infinite families of “subordinated” structures of lower order, including Poisson bracket structure, which satisfy certain matching conditions. The notion of Nambu bracket enables us to define Nambu-Poisson manifolds—phase spaces for the Nambu mechanics, which turn out to be more “rigid” than Poisson manifolds—phase spaces for the Hamiltonian mechanics. We introduce the analog of the action form and the action principle for the Nambu mechanics. In its formulation, dynamics of loops (n−2-dimensional chains for the generaln-ary case) naturally appears. We discuss several approaches to the quantization of Nambu mechanics, based on the deformation theory, path integral formulation and on Nambu-Heisenberg “commutation” relations. In the latter formalism we present an explicit representation of the Nambu-Heisenberg relation in then=3 case. We emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from Hamilton's to Nambu's dynamical picture.