Nambu mechanics

Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase nambu mechanics and multiple Hamiltonians. In a previous paper [Prog.

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. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique. This is a preview of subscription content, log in via an institution to check access. Rent this article via DeepDyve. Institutional subscriptions. Google Scholar.

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. Download as PDF Printable version. Generalization of Hamiltonian mechanics involving multiple Hamiltonians.

E0 Gravity. C00 Quantum chromodynamics. F Experimental Astrophysics.

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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. 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. However, we can clearly see from his Acknowledgment in the paper that the generalization of Hamiltonian dynamics attempted in this work had been a theme to which he had been devoting himself for more than twenty years, since his early years at Osaka City University. He expressed his gratitude to K.

Nambu mechanics

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. These structures are realizations in quantum or classical multi-body systems and are governed by Hamiltonian mechanical systems that are intrinsic to their hierarchies.

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B69 Other topics in strong interactions and related phenomena. C32 Experiments using neutrino beams. D41 Nuclear matter aspects in nuclear astrophysics. E25 Stellar structure and evolution. I20 Liquid and solid helium. Open in new tab Download slide. E34 Accretion, accretion disks. Dynamical symmetries and Nambu mechanics. J11 Incompressible fluids. J35 Supercooled liquids and glasses. Yoneya T. Then, Eq. E3 Compact objects? C22 Electron-proton collider experiments. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical systems.

In mathematics , Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians.

B58 Supersymmetric Standard Model. B32 Renormalization and renormalization group equation. In mathematics , Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Since the fundamental identity would play a similar role to the Jacobi identity in the Hamiltonian dynamics, its violation implies that it would be difficult to formulate the Nambu statistical mechanics or quantize the Nambu mechanics. I30 Surfaces, interfaces and thin films. Email alerts Article activity alert. B87 Other topics in mathematical methods. The calculated trajectories of the quantum, Nambu, and classical mechanics are shown in Fig. E24 Star formation. E76 Quantum field theory on curved space. J73 Environmental physics. A43 Quantum spins.