Nambu Structures Research Articles

Relative logarithmic cohomology and Nambu structures of maximal degree

We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibres of the restriction of the function on the degeneracy hypersurface of the Nambu structure, inside the Milnor fibres of the function itself.

Deformation of singular foliations, 1: Local deformation cohomology

In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.

Hidden Nambu mechanics II: Quantum/semiclassical dynamics

Abstract Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. 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. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. 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.

Nambu structures on Lie algebroids and their modular classes

We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu–Poisson manifold M with its Nambu tensor $$\Lambda $$ as the modular class of the tangent Lie algebroid TM with Nambu structure $$\Lambda $$ . We show that many known properties of the modular class of a Nambu–Poisson manifold and that of a Lie algebroid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu–Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modular classes are closely related to Evens–Lu–Weinstein modular classes of Lie algebroids.

On the Volume Elements of a Manifold with Transverse Zeroes

Moser proved in 1965 in his seminal paper that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the set of critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption ($b$-Poisson structures). We do this using the desingularization technique introduced by Guillemin-Miranda-Weitsman and extend it to $b^m$-Nambu structures.

Nambu structures and associated bialgebroids

We investigate higher-order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu–Poisson manifold (of order $$n>2$$ ) gives rise to a special bialgebroid structure which is referred to as a weak Lie–Filippov bialgebroid (of order n). It is further demonstrated that such bialgebroids canonically induce a Nambu–Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie–Filippov bialgebroid over a point.

Nambu structures on four-dimensional real Lie groups and related superintegrable systems

We determine all third- and fourth-order Nambu tensors (Nambu structures) on four-dimensional real Lie groups. In addition, we obtain superintegrable systems using the fourth-order Nambu structures on some of these Lie groups as phase spaces with the symmetry groups A4,8 and A4,10.

New results on the linearization of Nambu structures

In a paper with Jean-Paul Dufour in 1999 [4], we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in Dufour (1999) [4], namely the case of smooth linearization of Nambu structures with a Type 1 hyperbolic linear part which satisfies a natural signature condition. In this paper, we will show that such hyperbolic Nambu structures are also smoothly linearizable. We will also give a strong version of the analytic linearization theorem in the analytic case, improving a result obtained in Dufour (1999) [4].

Nambu sigma model and effective membrane actions

We propose an effective action for a p′-brane with open p-branes ending on it. The action has dual descriptions similar to the commutative and non-commutative ones of the DBI action for D-branes and open strings. The Poisson structure governing the non-commutativity of the D-brane is replaced by a Nambu structure and the open–closed string relations are generalized to the case of p-branes utilizing a novel Nambu sigma model description of p-branes. In the case of an M5-brane our action interpolates between M5-actions already proposed in the literature and matrix-model like actions involving Nambu structures.

Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets

We demonstrate that elementary biochemical reactions defined by mass-action kinetics satisfy a particular Nambu structure. To this end, we express biochemical reaction equations in terms of Nambu brackets and certain ω-factors. The ω-factors account for the fact that mass-action kinetics exhibits in general flow fields with finite divergence. The proposed approach by means of Nambu brackets and ω-factors unifies divergence freeflow fields of Newtonian mechanics and flow fields with finite divergence of mass-action kinetics.

Structure of magnetic field lines

In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines.

ON PLANAR AND NON-PLANAR ISOCHRONOUS SYSTEMS AND POISSON STRUCTURES

We construct certain new classes of isochronous dynamical systems based on the recent constructions of Calogero and Leyvraz. We show how a Poisson structure can be ascribed to such equations in ℝ3 and indicate their connection with the Nambu structures.

Momentum Map and Action-Angle Variables for Nambu Dynamics

Momentum map is a reduction procedure that reduces the dimension of a Hamiltonian system to the lower ones. It is shown that behavior of the action-angle variables under the momentum map generates the new action-angle variables for the reduced system considered as a Nambu structure. The symmetrical top is given as an illustration.

Linearization of Nambu Structures

Nambu structures are a generalization of Poisson structures in Hamiltonian dynamics, and it has been shown recently by several authors that, outside singular points, these structures are locally an exterior product of commuting vector fields. Nambu structures also give rise to co-Nambu differential forms, which are a natural generalization of integrable 1-forms to higher orders. This work is devoted to the study of Nambu tensors and co-Nambu forms near singular points. In particular, we give a classification of linear Nambu structures (integral finite-dimensional Nambu-Lie algebras), and a linearization of Nambu tensors and co-Nambu forms, under the nondegeneracy condition.

Some remarks concerning Nambu mechanics

The structure of Nambu-Poisson brackets is studied and we establish that any Nambu tensor is decomposable. We show that every Nambu-Poisson manifold admits a local foliation by canonical Nambu-Poisson manifolds. Finally, a cohomology for Nambu (Lie) algebras which is adapted to the study of formal deformations of Nambu structures is introduced.