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 nambu mechanics become the so-called generalized Hamiltonians, 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 ].

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.

I79 Other topics. I0 Structure, mechanical and acoustical properties.

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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.

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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B82 Noncommutative geometry. D15 Mesic nuclei and exotic atoms. Consider a one-dimensional quantum system of two oscillators whose Hamiltonian is given by. Quantizing Nambu dynamics leads to intriguing structures [5] that coincide with conventional quantization ones when superintegrable systems are involved—as they must. B43 Models with extra dimensions. B85 Bethe ansatz, exact S-matrix. I79 Other topics. E 97 , B34 Field theories in lower dimensions. C33 Experiments using photon beams. B64 Lattice QCD. B3 Quantum field theory. I74 Magnetism in nanosystems. Feldmeier H.

We review some aspects of Nambu mechanics on the basis of works previously published separately by the present author.

The resulting variational equations are semiclassical equations which have the same forms as the Hamilton equations of motion in Eq. F41 Laboratory experiments. This work might provide guidance for formulating a statistical theory of Nambu mechanics, and our formalism presented in this article might provide example systems suitable for Nambu statistical mechanics to be tested. F31 Expectation and estimation of gravitational radiation. E 97 , A13 Other topics in mathematical physics. Hidden categories: Articles with short description Short description matches Wikidata. E30 Compact objects in general. Here we present two examples; one is an example of exact quantum dynamics and the other is semiclassical. I62 Vortices. F4 Dark matter, dark energy and particle physics. J40 Liquid crystals. E Theoretical Astrophysics and Cosmology. We compare the results with the corresponding quantum and classical results. Copy to clipboard.