Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. As required by the formulation of Nambu dynamics, the integrals of motion nambu mechanics these systems necessarily become the so-called generalized Hamiltonians.

Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon—Heiles model of two interacting oscillators. In , Nambu proposed a generalization of the classical Hamiltonian dynamics [ 1 ] that is nowadays referred to as the Nambu mechanics. The structure of Nambu mechanics has impressed many authors, who have reported studies on its fundamental properties and possible applications, including quantization of the Nambu bracket [ 2 — 12 ].

Nambu mechanics

In mathematics , Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In , Yoichiro Nambu suggested a generalization involving Nambu—Poisson manifolds with more than one Hamiltonian. The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. Conserved quantity characterizing a superintegrable system that evolves in N -dimensional phase space. Nambu mechanics can be extended to fluid dynamics, where the resulting Nambu brackets are non-canonical and the Hamiltonians are identified with the Casimir of the system, such as enstrophy or helicity. Quantizing Nambu dynamics leads to intriguing structures [5] that coincide with conventional quantization ones when superintegrable systems are involved—as they must. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools.

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We review some aspects of Nambu mechanics on the basis of works previously published separately by the present author. We try to elucidate the basic ideas, most of which were rooted in more or less the same ground, and to explain the motivations behind these works from a unified and vantage viewpoint. Various unsolved questions are mentioned. I would like to start this review 1 by first presenting a brief comment on the historical genesis of our subject. His other seminal works, such as those on a dynamical model of elementary particles based on an analogy with the BCS theory of superconductivity, the discovery of the string interpretation of the Veneziano amplitude, and many other notable works, were all generated under close interactions with the environment of the contemporary developments in physics of those periods. This is evidenced by the fact that in these cases more or less similar works by other authors appeared independently and almost simultaneously.

Nambu mechanics

We outline basic principles of a canonical formalism for the Nambu mechanics—a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in We introduce the analog of the action form and the action principle for the Nambu mechanics. 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.

Genshin vore

Google Scholar. E-mail: horikosi tcu. The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. F23 Neutrino mass, , mixing, oscillation and interaction. E2 Stars and stellar systems. G Beam Physics. The Poisson bracket should satisfy the Jacobi identity,. H22 Instrumentation for space observatory. A4 Statistical mechanics - equilibrium systems. G04 Beam sources.

In mathematics , Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold.

As shown in the next section, we must express the Poisson brackets in the consistency condition of Eq. This equation gives the exact quantum dynamics, and we can consider several approximated dynamics. These equations are equivalent to the Hamilton equations of motion in Eq. A57 Nonequilibrium steady states. E21 The sun and solar system. Download references. C33 Experiments using photon beams. Awata H. D14 Hypernuclei. F20 Instrumentation and technique. Although the Nambu results slightly deviate from the quantum results as time increases, they can reproduce the energy exchange between two oscillators. E50 Large scale structure in general. D24 Photon and lepton reactions. H34 Data acquisition. A01 Electromagentism.