[Paper Review] On Foundation of the Generalized Nambu Mechanics (Second Version)

Leon A. Takhtajan|arXiv (Cornell University)|Jan 1, 2008

Homotopy and Cohomology in Algebraic Topology9 references33 citations

TL;DR

This paper establishes the foundational framework for generalized Nambu mechanics by introducing the Nambu bracket—a higher-order analog of the Poisson bracket for n ≥ 3 observables. It formulates Nambu-Hamilton equations, proves the fundamental identity as a consistency condition, and introduces Nambu-Poisson manifolds, which exhibit greater rigidity than Poisson manifolds, while also developing an action principle involving loop dynamics and discussing quantization approaches.

ABSTRACT

We outline basic principles of canonical formalism for the Nambu mechanics—a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of Nambu bracket, which generalizes the Poisson bracket— a “binary” operation on classical observables on the phase space, to the “multiple” operation of higher order n ≥ 3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion—a system of ODE’s which involves n − 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 an 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 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 general n-ary case) naturally appears. We discuss several approaches to the quantization of

Motivation & Objective

• To formalize the canonical structure of Nambu mechanics as a generalization of Hamiltonian mechanics beyond the binary Poisson bracket.
• To identify the fundamental identity as a consistency condition ensuring the closure of the Nambu bracket for n ≥ 3 observables.
• To demonstrate that Nambu bracket structures generate an infinite hierarchy of lower-order compatible structures, including Poisson brackets, under matching conditions.
• To define Nambu-Poisson manifolds as phase spaces for Nambu dynamics and analyze their geometric rigidity compared to Poisson manifolds.
• To develop an action principle for Nambu mechanics involving the dynamics of (n−2)-dimensional chains (loops), extending classical variational principles.

Proposed method

• Introduce the Nambu bracket as a multilinear, skew-symmetric n-ary operation on classical observables, generalizing the Poisson bracket.
• Derive the Nambu-Hamilton equations of motion involving n−1 Hamiltonians, forming a system of ODEs governing phase space flow.
• Establish the fundamental identity for the Nambu bracket as a necessary condition for consistency, generalizing the Jacobi identity.
• Show that the Nambu bracket induces a hierarchy of subordinated structures, including Poisson brackets, satisfying compatibility conditions.
• Define Nambu-Poisson manifolds as smooth manifolds equipped with a Nambu bracket, exhibiting stronger geometric constraints than Poisson manifolds.
• Propose an analog of the action form and action principle for Nambu mechanics, where dynamics naturally involves loop-like (n−2)-dimensional chains.

Experimental results

Research questions

• RQ1 How can the Nambu bracket be consistently defined as a higher-order generalization of the Poisson bracket for n ≥ 3?
• RQ2 What algebraic identity ensures the consistency of the Nambu dynamics, and how does it generalize the Jacobi identity?
• RQ3 How do Nambu-Poisson manifolds differ geometrically from Poisson manifolds, and what constraints do they impose?
• RQ4 In what way does the Nambu bracket generate a hierarchy of lower-order structures such as Poisson brackets?
• RQ5 How can a variational principle be formulated for Nambu mechanics, and what role do (n−2)-dimensional chains play in this formulation?

Key findings

• The Nambu bracket satisfies a fundamental identity that generalizes the Jacobi identity and ensures the consistency of the Nambu-Hamilton equations of motion.
• The Nambu bracket structure induces an infinite hierarchy of compatible lower-order structures, including Poisson brackets, with specific matching conditions between them.
• Nambu-Poisson manifolds are geometrically more rigid than Poisson manifolds due to the stronger constraints imposed by the n-ary bracket.
• The action principle for Nambu mechanics naturally involves the dynamics of (n−2)-dimensional chains, generalizing the point-particle action to extended objects.
• The framework provides a foundation for quantization of Nambu mechanics, though the paper stops short of completing the quantization program.