Symmetries,integrating factors and Nambu mechanics

We show that if an n-dimensional autonomous dynamical system (DS) with a vector field (VF) which has constant divergence possesses n − 1 independent first integrals, then it admits a symmetry VF which involves Nambu mechanics (NM). If the DS is conservative, then the Nambu VF happens to be a symmetry VF of the DS. We also show that the integrating factors can be constructed via NM. We illustrate our results on the Lotka-Volterra DS.     

Dynamical symmetries and 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. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.     

On foundation of the generalized Nambu mechanics

We outline basic principles of a canonical formalism for the Nambu mechanics—a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of a Nambu bracket, which generalizes the Poisson bracket—a binary operation on classical observables on the phase space—to the multiple operation of higher ordern3. Nambu dynamics is described by the phase flow given by Nambu-Hamilton equations of motion—a system of ODE's which involvesn–1 Hamiltonians. We introduce the fundamental identity for the Nambu bracket—a generalization of the Jacobi identity—as a consistency condition for the dynamics. We show that Nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order, including Poisson bracket structure, which satisfy certain matching conditions. The notion of Nambu bracket enables us to define Nambu-Poisson manifolds—phase spaces for the Nambu mechanics, which turn out to be more rigid than Poisson manifolds—phase spaces for the Hamiltonian mechanics. We introduce the analog of the action form and the action principle for the Nambu mechanics. In its formulation, dynamics of loops (n–2-dimensional chains for the generaln-ary case) naturally appears. We discuss several approaches to the quantization of Nambu mechanics, based on the deformation theory, path integral formulation and on Nambu-Heisenberg commutation relations. In the latter formalism we present an explicit representation of the Nambu-Heisenberg relation in then=3 case. 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.     

Algebraic structure of Nambu mechanics

Abstracting from Nambu’s work [1] on the generalization of Hamiltonian mechanics, we obtain the concept of a classical Nambu algebra of type I (CNA-I). Consistency requirement of time evolution of the trilinear Nambu bracket leads to a new five point identity (FPI). Incorporating the FPI into CNA-I, we obtain a classical Nambu algebra of type II (CNA-II). Nambu’s algorithm for generalized classical mechanics turns out to be compatible with CNA-II. Tensor product composition of two CNA-I’s results in another CNA-I whereas that of two CNA-II’s does not. This implies that interacting systems cannot be consistently treated in Nambu’s framework. It is shown that the recent generalization of Nambu mechanics based on an arbitrary Lie group (instead of the particular case of the rotation group as in the case of Nambu’s original algorithm) suggested by Biyalinicki-Birula and Morrison [2], is compatible with CNA-I but not with CNA-II. Relaxation of the commutative and associative observable product by making it nonassociative so as to arrive at the quantum counterpart meets with serious difficulties from the view point of tensor product composition property. Thus neither CNA-I nor CNA-II have quantum counterparts. Implications of our results are discussed with special reference to existing work on Nambu mechanics in the literature.     

Generalization of Nambu's mechanics

We look for a generalization of the mechanics of Hamilton and Nambu. We have found the equations of motion of a classical physical system ofS basic dynamic variables characterized byS – 1 constants of motion and by a function of the dynamical variables and the time whose value also remains constant during the evolution of the system. The numberS may be even or odd. We find that any locally invertible transformations are canonical transformations. We show that the equations of motion obtained can be put in a form similar to Nambu's equations by means of a time transformation. We study the relationship of the present formalism to Hamiltonian mechanics and consider an extension of the formalism to field theory.     

Phase space action for minimal surfaces of any dimension in curved spacetime

The phase space action, depending on coordinates, momenta and Lagrange multipliers (which turn out to be components of the induced metric), has been written down for a world sheet of arbitary dimension (a string generalized to membranes) in a curved embedding spacetime. Canonical and hamiltonian formalisms have been formulated in a covariant and general way with the true dynamical variables being separated from the redundant ones. The membrane constraints follow directly from the variation of our action; their suitable superposition gives a hamiltonian from which we derive the equations of motion for a membrane via the Poisson brackets. The same hamiltonian we obtain also in a different way from a variation of the action. For n = 2 all equations coincide with those of strings.     

A Hamilton-Jacobi formalism for thermodynamics

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it is 2n+1-dimensional. The equations of state of a substance pick out an n-dimensional submanifold. A family of substances whose equations of state depend on n parameters define a hypersurface of co-dimension one. This can be described by the vanishing of a function which plays the role of a Hamiltonian. The ordinary differential equations (characteristic equations) defined by this function describe a dynamical system on the hypersurface. Its orbits can be used to reconstruct the equations of state. The ‘time’ variable associated to this dynamics is related to, but is not identical to, entropy. After developing this formalism on well-grounded systems such as the van der Waals gases and the Curie-Weiss magnets, we derive a Hamilton-Jacobi equation for black hole thermodynamics in General Relativity. The cosmological constant appears as a constant of integration in this picture.     

Topological Field Theory Interpretation of String Topology

The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S¹-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n,) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.     

Generalized Flinn operators with applications to multi-component alloys

The relationship between the occupation operator, spin operator and Flinn operator formalisms for a substitutional binary alloy is presented. These three methods are generalized to multicomponent alloys with irreducible static n-body potentials. In this manner the concept of generalized Flinn operators is introduced. The configurational Hamiltonians and generalized Flinn operators are then explicitly presented for ternary and quaternary alloys with static two-body potentials. The merits of the generalized Flinn operator approach are discussed.     

The electron field and the Dirac bracket

A recent quantization rule of Fermi systems starts from the new symmetric brackets of classical mechanics. As a consequence, Fermi and Bose quantization can be put on an equal footing, instead of the standardad hoc procedure. We prove that the rule gives the right anticommutation relations when applied to the case of the relativistic electron. We show that this is a crucial test of the rule.For completeness, Dirac's Hamiltonian mechanics and the plus and minus Dirac bracket formalisms are developed for the electron's field.     

Algebraic integrability and generalized symmetries of dynamical systems

It has been shown that when an n-dimensional dynamical system admits a generalized symmetry vector field which involves a divergence-free Liouville vector field, then it possesses n−1 independent first integrals (i.e., it is algebraically integrable). Furthermore, the Liouville vector field can be employed for the classification of algebraically integrable dynamical systems. The results have been discussed on examples which arise in physics.     

k-Cosymplectic Classical Field Theories: Tulczyjew and Skinner–Rusk Formulations

The k-cosymplectic Lagrangian and Hamiltonian formalisms of first-order classical field theories are reviewed and completed. In particular, they are stated for singular and almost-regular systems. Subsequently, several alternative formulations for k-cosymplectic first-order field theories are developed: First, generalizing the construction of Tulczyjew for mechanics, we give a new interpretation of the classical field equations. Second, the Lagrangian and Hamiltonian formalisms are unified by giving an extension of the Skinner–Rusk formulation on classical mechanics.     

NAMBU BRACKET FORMULATION OF NONLINEAR BIOCHEMICAL REACTIONS BEYOND ELEMENTARY MASS ACTION KINETICS

We develop a Nambu bracket formulation for a wide class of nonlinear biochemical reactions by exploiting previous work that focused on elementary biochemical mass action reactions. To this end, we consider general reaction mechanisms including for example enzyme kinetics. Furthermore, we go beyond elementary reactions and account for reactions involving stoichiometric coefficients different from unity. In particular, we show that the stoichiometric matrix of biochemical reactions can be expressed in terms of Nambu brackets. Finally, we solve the sign problem that arises in the context of coupled biochemical reactions.     

Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an ‘n-plectic manifold’: a finite-dimensional manifold equipped with a closed nondegenerate (n + 1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.     

Volume Preserving Multidimensional Integrable Systems and Nambu–Poisson Geometry

Abstract

In this paper we study generalized classes of volume preserving multidimensional integrable systems via Nambu–Poisson mechanics. These integrable systems belong to the same class of dispersionless KP type equation. Hence they bear a close resemblance to the self dual Einstein equation. All these dispersionless KP and dToda type equations can be studied via twistor geometry, by using the method of Gindikin’s pencil of two forms. Following this approach we study the twistor construction of our volume preserving systems.     

Applications of Nambu Mechanics to Systems of Hydrodynamical Type II

Abstract

In this paper we further investigate some applications of Nambu mechanics in hydrodynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189–L1193] had demonstrated the connection between Nambu mechanics and noncanonical Hamiltonian mechanics. Nambu mechanics is extended to incompressible ideal hydrodynamical fields using energy and helicity in three dimensional (enstrophy in two dimensional). In this paper we discuss the Lax representation of systems of Névir-Blender type. We also formulate the three dimensional Euler equations of incompressible fluid in terms of Nambu-Poisson geometry. We discuss their Lax representation. We also briefly discuss the Lax representation of ideal incompressible magnetohydrodynamics equations.     

Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body

Abstract

In this work we investigate a formal mapping between the dynamical properties of the unidimensional relativistic oscillator and the asymmetrical rigid top at a classical level. We study the relativistic oscillator within Yamaleev's interpretation of Nambu mechanics. Such interpretation is based on the factorisation of the momenta, and as a consequence of this factorisation we are led to a three dimensional phase space. Solutions of the relativistic oscillator are given in terms of the Jacobian elliptic functions and hence we establish a correspondence of these solutions in terms of well known quantities from the rigid body theory. We also study some mechanical restrictions that appear in the mathematical development of the mapping. In particular, we find a lower bound for the relativistic frequency in order to make the mapping self-consistent and physically legitimate.