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 nambu mechanics 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. We derived the consistency condition to determine the induced constraints. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical systems. The key idea is as follows. Furthermore, if these constraints are constants of motion, the time evolution of the Nambu multiplet could be given by the Nambu equations.

Provided by the Springer Nature SharedIt content-sharing initiative. Therefore the nambu mechanics of two oscillators considered here is anomalous as the Nambu mechanics, but not anomalous as the Hamiltonian dynamics. B29 Other topics in string theory.

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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

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.

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F2 Neutrino. I79 Other topics. H02 Simulation and detector modeling. Mongkolsakulvong S. B62 Hadronic colliders. The second model is a quantum system which exhibits nonlinear energy exchange dynamics between coupled oscillators. Horikoshi, in preparation. Then, Eq. Heller E. A3 Nonlinear dynamics. These equations are equivalent to the Hamilton equations of motion in Eq. Science and Mathematics.

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.

E53 Large scale structure formation. G00 Colliders. Anyone you share the following link with will be able to read this content:. A systematic approximation scheme to derive higher-order semiclassical dynamics, the quantized Hamiltonian dynamics [ 16 , 17 ], has been developed. C34 Other topics. H50 Detector system design, construction technologies and materials. B Theoretical Particle Physics. J28 Non-neutral plasma, dust plasma. Citing articles via Web of Science 2. Thermodynamic topology of black holes in f R gravity. E Theoretical Astrophysics and Cosmology. H10 Experimental detector systems. I2 Quantum fluids and solids. B44 Technicolor and composite models. A31 The other dynamical systems such as cellular-automata and coupled map lattices.