相关论文: Linearization of Nambu Structures

We present recent developments in the theory of Nambu mechanics, which include new examples of Nambu-Poisson manifolds with linear Nambu brackets and new representations of Nambu-Heisenberg commutation relations.

The theory of Nambu-Poisson structures on manifolds is extended to the context of Lie algebroids, in a natural way based on the Vinogradov bracket associated with Lie algebroid cohomology. We show that, under certain assumptions, any…

Taking as a model the fact that Heisenberg's matrix mechanics was derived from Hamiltonian mechanics using the correspondence principle, we explore a class of dynamical systems involving discrete variables, with Nambu mechanics as the…

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a…

Integrable deformations of a class of Rikitake dynamical systems are constructed by deforming their underlying Lie-Poisson Hamiltonian structures, which are considered linearizations of Poisson--Lie structures on certain (dual) Lie groups.…

Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions…

We study hyper-elliptic Nambu flows associated with some $n$ dimensional maps and show that discrete integrable systems can be reproduced as flows of this class.

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the…

Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is…

We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable…

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered.…

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $\Lambda$ as the modular class of the tangent…

We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), that satisfy particular…

In this paper we revisit the classification of the gauge transformations in the Euler top system using the generalized classical Hamiltonian dynamics of Nambu. In this framework the Euler equations of motion are bi-Hamiltonian and $SL(2,…

We have determined all Nambu tensors (Nambu structures) of order four and three on four dimensional real Lie groups. Furthermore, we have obtained superintegrable systems by use of the Nambu structures of order four on some of these Lie…

Takhtajan has recently studied the consistency conditions for Nambu brackets, and suggested that they have to be skew-symmetric, and satisfy Leibnitz rule and the Fundamental Identity (FI, it is a generalization of the Jacobi identity). If…

Using the notion of a contravariant derivative, we give some algebraic and geometric characterizations of Poisson algebras associated to the infinitesimal data of Poisson submanifolds. We show that such a class of Poisson algebras provides…

In this paper we introduce a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra's system. We found in two simplest cases the complete sets of the integrals of motion using Nambu--Poisson…

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…