Nambu mechanics

It is shown nambu mechanics several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized 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.

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.

These equations are equivalent to semiclassical equations of motion derived from the time-dependent variational principle of Eq.

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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.

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.

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J11 Incompressible fluids. B4 Model building. H1 Detectors, apparatus and methods for the physics using accelerators. In many-degrees-of-freedom systems, however, the hidden Nambu mechanics become anomalous, because interactions between multiple degrees of freedom violate the fundamental identity of Eq. H11 Gaseous detectors. Toggle limited content width. Hidden Nambu mechanics. More metrics information. In a previous paper [Prog. Therefore the dynamics of two oscillators considered here is anomalous as the Nambu mechanics, but not anomalous as the Hamiltonian dynamics. I32 Graphene, fullerene. F23 Neutrino mass, , mixing, oscillation and interaction. C01 Electroweak model, Higgs bosons, electroweak symmetry breaking. A00 Classical mechanics. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians.

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.

C31 Experiments using charged lepton beams. Here we make some comments on the functional forms of Nambu Hamiltonians. A41 Spin-glass, random spins. F03 Ultra-high energy phenomena of cosmic rays. On the other hand, due to the lack of the zero-point energy effect, the classical results are very different from the quantum mechanical results. F12 Radiation from extragalactic objects. Modin K. Without the Jacobi identity the canonical transformation of the canonical doublet cannot be properly defined, and the dynamics becomes anomalous [ 15 ]. C21 Lepton collider experiments. We derived the consistency condition to determine the induced constraints. A61 Quantum information quantum computation, quantum cryptography, quantum communication etc. Citing articles via Web of Science 2. J32 Solutions and liquids. G03 Electron accelerators.