Nambu Mechanics Research Articles

Fractional Nambu dynamics

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Nambu system with fractional derivatives (the fractional Nambu dynamics), and study its applications to mechanical systems. By using the definition of combined fractional derivative, we construct unified fractional Nambu equations and, respectively, give four new kinds of fractional Nambu equations based on the different definitions of fractional derivative. Further, we study the relations between the fractional Nambu system, the fractional generalized Hamiltonian system, the fractional Birkhoffian system, the fractional Hamiltonian system, the fractional Lagrangian system and a series of integer-order dynamical systems and present the transformation conditions. And furthermore, as applications of the fractional Nambu method, we construct two kinds of fractional dynamical models, which include the fractional Euler–Poinsot model of rigid body which rotates with respect to a fixed point and the fractional Yamaleev oscillator model. This work provides a general method for studying a fractional dynamical problem which is related to science and engineering.

Hydrodynamics on non-commutative space: A step toward hydrodynamics of granular materials

Hydrodynamics on non-commutative space is studied based on a formulation of hydrodynamics by Y. Nambu in terms of Poisson and Nambu brackets. Replacing these brackets by Moyal brackets with a parameter $\theta$, a new hydrodynamics on non-commutative space is derived. It may be a step toward to find the hydrodynamics of granular materials whose minimum volume is given by $\theta$. To clarify this minimum volume, path integral quantization and uncertainty relation of Nambu dynamics are examined.

A geometric application of Nambu mechanics: the motion of three point vortices in the plane

In this paper the dynamics of three point vortices in the plane is analysed in terms of Nambu mechanics and compared to the classical Hamiltonian dynamics. Two new aspects are introduced: (i) the motion of three point vortices can be classified as non-canonical Nambu mechanics and (ii) Nambu mechanics leads to a geometric representation of the trajectories in an adapted three-dimensional phase space without explicitly solving the differential equations. Thereby, the point vortex trajectory is given by the intersection of two conserved quantities as surfaces in the phase space. These constitutive quantities are the total energy and an angular momentum based Casimir function of the dynamical system. The topological structure of the last surface represents a one- or two- sheeted hyperboloid, a cone or an ellipsoid in the phase space. Examples of the periodic motion and a novel aspect of the collapse of three point vortices in the unbounded plane are discussed. Furthermore, an approach to generalize Nambu mechanics for an arbitrary number of point vortices is proposed.

Dissipative Nambu systems and oscillator circuit design

Received January 5, 2014; Published July 1, 2014Abstract: Electronic oscillators have to be described by nonlinear differential equations witha limit cycle where analytical solutions are rarely predictable. Therefore, design concepts ofoscillators are not available where a designer can start with a complete set of specifications ofthe desired oscillator. Only linear aspects of the oscillator design can be solved in a reasonablemanner. In this paper we present a mathematical concept that is appropriate to constructionnonlinear differential equations with a limit cycle. For this purpose we use a concept fromtheoretical physics, the so-called Nambu mechanics, and add canonical dissipation. It can beshown that a limit cycle arises under certain conditions. Based on the corresponding systemof differential equations a SIMULINK model is constructed that can be the basis of a circuitrealization.Key Words: nonlinearcircuits, oscillators, dynamicalsystems, limitcycles, Nambumechanics,canonical dissipation

On symmetries and conserved quantities in Nambu mechanics

In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number. The origin of the difference may be traced back to a shift in degrees of relevant forms present in equations of motion, or, alternatively, to a corresponding shift in degrees of relevant objects in action integral for Nambu mechanics.

Hidden Nambu mechanics: A variant formulation of Hamiltonian systems

We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.

Infinite-dimensional 3-algebra and integrable system

The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV hierarchy follows from the Nambu-Poisson evolution equation given the suitable Hamiltonians. We find that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. Due to the Nambu-Poisson evolution equation involving two Hamiltonians, more intriguing relationships between these Hamiltonians are revealed. As an application, we investigate the system of polytropic gas equations and derive an integrable gas dynamics system in the framework of Nambu mechanics.

MATRIX QUANTIZATION OF TURBULENCE

Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos 2009], we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of noncommutative phase space coordinates as Hermitian N × N matrices in R3. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction, it violates the quantum commutation relations. We demonstrate that the Heisenberg–Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand, there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving nondissipative sector survive for long times.

Phase space geometry and chaotic attractors in dissipative Nambu mechanics

Following the Nambu mechanics framework, we demonstrate that the non-dissipative part of the Lorenz system can be generated by the intersection of two quadratic surfaces that form a doublet under the group . All manifolds are classified into four distinct classes: parabolic, elliptical, cylindrical and hyperbolic. The Lorenz attractor is localized by a specific infinite set of one-parameter family of these surfaces. The different classes correspond to different physical systems. The Lorenz system is identified as a charged rigid body in a uniform magnetic field with external torque and this system is generalized to give new strange attractors.

Vector Product and an Integrable Dynamical System

We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.

Canonical form of Nambu–Poisson bracket: A pedestrian approach

In the seventies, Nambu (Phys. Rev. D7, 2405 (1973)) proposed a new approach to classical dynamics based on an N -dimensional Nambu-Poisson (NP) manifold replacing the primi- tive even-dimensional Poisson manifold and on N -1 Hamiltonians in place of a single Hamiltonian. This approach has had many promoters including Bayen and Flato (Phys. Rev. D11, 3049 (1975)), Mukunda and Sudarshan (Phys. Rev. D13, 2846 (1976)), and Takhtajan (Comm. Math. Phys. 160, 295 (1994)) among others. While Nambu had originally considered N = 3, the illustration of his ideas for N = 4a nd 6w as given by Chatterjee ( Lett. Math. Phys. 36, 117 (1996)) who observed that the classical description of dynamical systems having dynamical symmetries is described ele- gantly by Nambu's formalism of mechanics. However, his considerations do not quite yield the beautiful canonical form conjectured by Nambu himself for the N -ary NP bracket. By making a judi- cious choice for the 'extra constant of motion' of namely, α and β, which are the orientation angles in Kepler problem and isotropic harmonic oscillator (HO) respectively, we show that the dynami- cal systems with dynamical symmetries can be recast in the beautiful form suggested by Nambu. We believe that the techniques used and the theorems suggested by us in this work are of gen- eral interest because of their involvement in the transition from Hamiltonian mechanics to Nambu mechanics.

Autonomous Dynamical Systems, Explicitly Time Dependent First Integrals, and Gradient Systems

Note that the system also admits the independent first integral (c12u1− c23u3)/2, and the dynamical system can be reconstructed using Nambu mechanics. If c 6= 0, then I1, I2 are not first integrals anymore. Now many dissipative dynamical systems such as the Lorenz model and the Rikitake-two disc dynamo admit, depending on the bifurcation parameters, explicitly time-dependent first integrals. These type of first integrals are of the form [3 – 9]

Phase flows and vector Hamiltonians

We present a generalization of the Nambu mechanics on the base of Liouville’s theorem. We prove that the Poisson structure of an n-dimensional multisymplectic phase space is induced by (n − 1)-Hamiltonian k-vector fields each of which requires introduction of k-Hamiltonians.

D1-brane in constant R-R 3-form flux and Nambu dynamics in string theory

We consider D1-string in a constant R-R 3-form flux background and analyze its low energy limit. The leading order low energy theory has reparametrization symmetry and is a generalization of an earlier work by Takhtajan. We show that the dynamical evolution of the theory takes a generalized Hamiltonian form in terms of a Nambu bracket. This description is formulated in terms of reparametrization invariant quantities and requires no fixing of the reparametrization symmetry. We also show that a Nambu-Poisson (p + 2)-bracket arises naturally in the reparametrization invariant description of the low energy theory of a p-brane in a constant (p + 2)-form flux background. For example, our results apply for a fundamental string in a constant NS-NS 3-form flux H 3 and an M2-brane in a constant 4-form flux F 4.

Fractional Odd-Dimensional Mechanics

The classical Nambu mechanics is generalized to involve fractional derivatives using two different methods. The first method is based on the definition of fractional exterior derivative and the second one is based on extending the standard velocities to the fractional ones. Fractional Nambu mechanics may be used for nonintegrable systems with memory. Further, Lagrangian which is generate fractional Nambu equations is defined.

A Fokker–Planck approach to canonical-dissipative Nambu systems: With an application to human motor control during dynamic haptic perception

We formulate Markov diffusion processes for canonical-dissipative systems exhibiting Nambu mechanics. Analytical expressions for stationary canonical-dissipative distributions are obtained. Nambu–Boltzmann distributions are derived as special cases for systems without energy pumping. The Markov short-time propagator is used to derive maximum likelihood estimators for parameters of a model that describes a particular dynamic motor pattern providing haptic cues.

ON TWO HIERARCHIES OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS, FIRST INTEGRALS, NAMBU MECHANICS, AND PAINLEVÉ PROPERTY

We study a hierarchy of nonlinear autonomous systems of first ordinary differential equations. We show that it is completely integrable using the Painlevé test and finding the first integrals. We also show that it can be derived from the first integrals using Nambu mechanics. A corresponding dynamical system with chaotic behavior is also derived. A modified system with harmonic oscillators as first integrals is also considered. A connection with the Yang–Mills equation and the self-dual Yang–Mills equation is also discussed.

Strange attractors in dissipative Nambu mechanics: classical and quantum aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $ N \times N $ matrices in $ R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.

Active systems with Nambu dynamics: with applications to rod wielding for haptic length perception and self-propagating systems on two-spheres

We generalize Nambu dynamics to describe self-excited, active systems that evolve on attractors and account for the up-take of energy and activation processes, on the one hand, and damping and inhibitory processes, on the other. An application to rod wielding for haptic length perception and a model for self-propagating systems on two-spheres are discussed.

Nambu quantum mechanics on discrete 3-tori

We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus represented by elements of the group . These flows can be considered as special motions of the Nambu dynamics (linear Nambu flows) in the three-dimensional toroidal phase space and are characterized by invariant vectors a of . We quantize all such flows, which are necessarily restricted on a planar two-dimensional phase space, embedded in the 3-torus, transverse to the vector a. The corresponding maps belong to the little group of , which is an subgroup. The associated linear Nambu maps are generated by a pair of linear and quadratic Hamiltonians (Clebsch–Monge potentials of the flow) and the corresponding quantum maps realize the metaplectic representation of on the discrete group of three-dimensional magnetic translations, i.e. the non-commutative 3-torus with a deformation parameter the Nth root of unity. Other potential applications of our construction are related to the quantization of deterministic chaos in turbulent maps as well as to quantum tomography of three-dimensional objects.