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 for these systems necessarily become the so-called generalized Hamiltonians, 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.

Nambu mechanics

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 ]. However, the applications to date have been limited to particular systems, because Nambu systems generally require multiple conserved quantities as Hamiltonians and the Nambu bracket exhibits serious difficulties in systems with many degrees of freedom or quantization [ 1 , 2 , 11 ]. In we proposed a new approach to Nambu mechanics [ 13 ]. We revealed that the Nambu mechanical structure is hidden in a Hamiltonian system which has redundant degrees of freedom.

J35 Supercooled liquids and glasses.

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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. This is a preview of subscription content, log in via an institution to check access. Rent this article via DeepDyve. Institutional subscriptions.

Nambu mechanics

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. The case of GHD, in contrast, seems entirely different. As far as I know, he himself never mentioned this paper in his later research papers, except for some expository accounts or reminiscences.

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I97 Other topics. F1 Photons. C43 Underground experiments. B46 Other topics in model building. J0 Mechanics, elasticity and rheology. Advance article alerts. E21 The sun and solar system. AMS 52 , I9 Low dimensional systems -electronic properties. 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. Then, the conditions in Eq. F14 Cosmic microwave background and extragalactic background lights. I53 Thermal transport. F10 Instrumentation and technique.

Nambu mechanics [ 1 ] provides a means to view, in perspective, a diversity of phenomena from micro- to macro- and to cosmic-scales, with ordered structures characterized by helicity and chirality, and to approach the secret of their formations. The helicity was discovered for elementary particles, but the same terminology is given to an invariant for motion of a fluid. These structures are ubiquitous, but their formation process remains puzzles.

J22 Waves, heating, instabilities. I71 Frustration. The Nambu bracket of Eq. Issue Date : February E01 Relativity. J56 Other topics in biophysics. D00 Nuclear forces including two nucleon problems. J33 Quantum chemistry, electronic states. E13 Kinetic theory and plasma. Journal Article.