Nambu mechanics

Nambu mechanics is a generalized Nambu mechanics 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 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. 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.

A24 Other numerical methods. The lowest-order approximation is simply the classical Hamiltonian dynamics.

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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, Y.

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. Phenomena of meso- and macro-scales have infinite degrees of freedom and are described by partial differential equations. The Hamilton structure of many degrees of freedom is often degenerate. In case nonlinear terms are quadratic, the governing equation takes the form of the Lie-Poisson equation.

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As preparation for the next section, we give a detailed description of some examples. Here we present detailed descriptions of two simple examples to show how induced constraints are obtained for given multiplets. G22 Industrial application. D5 Other topics in nuclear physics. C41 Laser experiments. H11 Gaseous detectors. E36 Massive black holes. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique. I11 Thermal transport. D02 Weak interactions in nuclear system including neutrino-nuclear interactions.

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.

B05 Quantization and formalism. J20 Nuclear fusions. F14 Cosmic microwave background and extragalactic background lights. A11 Solitons. G02 Ion accelerators. As preparation for the next section, we give a detailed description of some examples. These equations are equivalent to semiclassical equations of motion derived from the time-dependent variational principle of Eq. That is, the Nambu structure is hidden in the semiclassical dynamics of the frozen Gaussian wave packet. B61 Deep inelastic scattering. B43 Models with extra dimensions. I11 Thermal transport. More metrics information. I8 Optical properties. F0 Cosmic ray particles.