On foundation of the generalized Nambu mechanics
type: Item
instance of: scholarly article
title: On foundation of the generalized Nambu mechanics
zbMATH Open document ID: 0808.70015
DOI: 10.1007/BF02103278
author: Leon A. Takhtajan
published in: Communications in Mathematical Physics
publication date: 1995-03-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9301111
review text: This article outlines basic principles of canonical formalism for Nambu mechanics (a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973). For instance, the canonical Nambu bracket for a triple of classical observables \(f_ i: \mathbb{R}^ 3\to \mathbb{R}\) is given by \(\{f_ 1, f_ 2, f_ 3\}= {{\partial(f_ 1, f_ 2, f_ 3)} \over {\partial(x_ 1, x_ 2, x_ 3)}}\). The following fundamental identity is satisfied: \[ \{\{f_ 1, f_ 2, f_ 3\},f_ 4, f_ 5\}+ \{f_ 3,\{f_ 1, f_ 2, f_ 4\},f_ 5\}+ \{f_ 3, f_ 4,\{f_ 1, f_ 2, f_ 5\}\}= \{f_ 1, f_ 2,\{f_ 3, f_ 4, f_ 5\}\}.\tag \(*\) \] This resembles Jacobi identity for the usual Poisson brackets. Besides, skew symmetry and Leibniz rule: \(\{f_ 1, f_ 2, f_ 3, f_ 4\}= f_ 1\{f_ 2, f_ 3, f_ 4\}+ f_ 2\{f_ 1, f_ 3, f_ 4\}\) are also satisfied. Equations of motion for given two ``Hamiltonians'' \(H_ 1\), \(H_ 2\) are the following, by definition: \(df/dt= \{H_ 1, H_ 2, f\}\). More generally, the author defines a Nambu-Poisson manifold of order \(n\) as a manifold \(X\) together with a map \(\{.,\dots\}: A^{\oplus^ n}\to A\), where \(A\) is the ring of \(C^ \infty\)-functions on \(X\), and where \(\{.,\dots\}\) is skew symmetric and satisfies Leibniz rule as well as identity \((*)\): \[ \begin{multlined} \{\{f_ 1,\dots, f_{n-1}\}, f_{n+1},\dots, f_{2n-1}\}+ \{f_ n, \{f_ 1,\dots, f_{n-1}, f_{n+1}\}, f_{n+2},\dots, f_{2n-1}\}+\cdots\\ +\{f_ n,\dots, f_{2n-2}, \{f_ 1,\dots, f_{n-1}, f_{2n-1}\}\}= \{f_ 1,\dots, f_{n-1}, \{f_ n,\dots, f_{2n-1}\}\}. \end{multlined} \] Then many basic concepts and properties of Hamiltonian mechanics are generalized in the context of Nambu mechanics (integrals of motion, action principle). Finally quantization procedures are also described (generalizing Feynman's path integral, and canonical approach to quantization). It is, in my opinion, a reasonably self-contained and interesting article.
Mathematics Subject Classification ID: 70H99
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 81S05
zbMATH DE Number: 529873
zbMATH Keywords: equations of motion
zbMATH Keywords: integrals of motion
zbMATH Keywords: canonical Nambu bracket
zbMATH Keywords: Leibniz rule
zbMATH Keywords: Nambu-Poisson manifold
zbMATH Keywords: action principle
zbMATH Keywords: Feynman's path integral
reviewed by: Hernan Cendra
MaRDI profile type: Publication
arXiv ID: hep-th/9301111
cites work: Generalized Hamiltonian Dynamics
cites work: Reductions of self-dual Yang-Mills fields and classical systems
cites work: Deformation theory and quantization. I: Deformations of symplectic structures
cites work: Algebras and triangle relations
cites work: Sur certains groupes d'opérateurs unitaires
description: scientific article