Poisson Bracket Formulation Research Articles

A new canonical affine bracket formulation of Hamiltonian classical field theories of first order

It has been a long standing question how to extend, in the finite-dimensional setting, the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang–Mills theory.

Poisson Bracket Formulation for a Dissipative Two- Dimensional Anisotropic Harmonic Oscillator with Fractional Derivatives: Analysis and Applications

Poisson Bracket Formulation for a Dissipative Two- Dimensional Anisotropic Harmonic Oscillator with Fractional Derivatives: Analysis and Applications

Poisson-bracket formulation of the dynamics of fluids of deformable particles.

Using the Poisson-bracket method, we derive continuum equations for a fluid of deformable particles in two dimensions. Particle shape is quantified in terms of two continuum fields: an anisotropy density field that captures the deformations of individual particles from regular shapes and a shape tensor density field that quantifies both particle elongation and nematic alignment of elongated shapes. We explicitly consider the example of a dense biological tissue as described by the Vertex model energy, where cell shape has been proposed as a structural order parameter for a liquid-solid transition. The hydrodynamic model of biological tissue proposed here captures the coupling of cell shape to flow and provides a starting point for modeling the rheology of dense tissue.

Generalizing Poisson Algebra with Geometry

Using a differential geometric interpretation of Hamiltonian mechanics, a generalized Poisson bracket formulation is developed for a three-dimensional phase space characterized by a triplet of canonical variables.

Poisson bracket formulation of a higher-order, geometrically-exact beam

This paper investigates the Hamiltonian structure and Poisson bracket formulation of a higher-order, geometrically-exact Cosserat type beam with a deforming cross-section in terms of canonically conjugate variables.

Local conservative regularizations of compressible magnetohydrodynamic and neutral flows

Ideal systems like magnetohydrodynamics (MHD) and Euler flow may develop singularities in vorticity (w=∇×v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper, we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length λ which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/ωpe. Our regularization preserves the symmetries of the original systems and, with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied to generalize the constitutive relation to bound higher moments of vorticity. A “swirl” velocity field is identified, and shown to transport w/ρ and B/ρ, generalizing the Kelvin-Helmholtz and Alfvén theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch, and a plane vortex sheet. The proposed regularization could facilitate numerical simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma dynamic systems arising from our work are briefly discussed.

Hamiltonian gyrokinetic Vlasov–Maxwell system

A new formulation of electromagnetic gyrokinetics that possesses Hamiltonian form is constructed. The new formulation replaces Poisson-like equations by hyperbolic equations for the electromagnetic field with the speed of light slowed to that of the gyrokinetic vacuum, thereby significantly reducing computational cost. An energy principle is derived using the field-theoretic noncanonical Poisson bracket formulation of the theory. The energy principle is used to prove stability of the thermal equilibrium state in a uniform background magnetic field.

Pseudo-Riemannian Geometry in Terms of Multi-Linear Brackets

We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)K\"ahler geometry.

A Lie–Poisson bracket formulation of plasticity and the computations based on the Lie-group [formula omitted

In this paper we develop a generalized Hamiltonian formulation of a perfectly elastoplastic model, which is a typical dissipative system. On the cotangent bundle of the yield manifold, a Lie–Poisson bracket is used to construct the differential equations system. The stress trajectory is a coadjoint orbit on the Poisson manifold under a coadjoint action by the Lie-group SO(n). The plastic differential equation is an affine non-linear system, of which a finite-dimensional Lie algebra can be constructed, and the superposition principle is available for this system. Accordingly, we can construct numerical schemes to automatically preserve the yield-surface for perfect plasticity, for isotropic hardening material, as well as for an anisotropic elastic–plastic model. Then, we describe an anisotropic elastic–plastic material model without entering the work-hardening range and deforming under a specified dissipation rate, which can be achieved through a stress-dependent feedback control law of strain rate.

Tracking energy transfer from excited to accepting modes: application to water bend vibrational relaxation

We extend, via a reformulation in terms of Poisson brackets, the method developed previously (Rey et al., J. Phys. Chem. A, 2009, 113, 8949) allowing analysis of the pathways of an excited molecule's ultrafast vibrational relaxation in terms of intramolecular and intermolecular contributions. In particular we show how to ascertain, through the computation of power and work, which portion of an initial excess molecular energy (e.g. vibrational) is transferred to various degrees of freedom (e.g. rotational, translational) of the excited molecule itself and its neighbors. The particular case of bend excess energy relaxation in pure water is treated in detail, completing the picture reported in the work cited above. It is shown explicitly, within a classical description, that almost all of the initial water bend excitation energy is transferred-either indirectly, via Fermi resonance centrifugal coupling to the bend-excited water's rotation, or directly via intermolecular coupling- to local water librations, only involving molecules in the first two hydration shells of the vibrationally excited water molecule. Finally, it is pointed out that the Poisson bracket formulation can also be applied to elucidate the microscopic character of solvation and rotational dynamics, and should prove useful in developing a quantum treatment for energy flow in condensed phases.

Hamiltonian–Dirac simulated annealing: Application to the calculation of vortex states

A simulated annealing method for calculating stationary states for models that describe continuous media is proposed. The method is based on the noncanonical Poisson bracket formulation of media, which is used to construct Dirac brackets with desired constraints, and symmetric brackets that cause relaxation with the desired constraints. The method is applied to two-dimensional vortex dynamics and a variety of numerical examples is given, including the calculation of monopole and dipole vortex states.

“Volume‐Preserving” Conformational Rheological Models for Multi‐Component Miscible Polymer Blends Using the GENERIC Formalism

AbstractA family of conformational rheological models for multi‐component miscible polymer blends is developed using a modified form of the Poisson bracket formulation. Two conformation tensors called c1 and c2 are introduced to show the orientation of the first and the second components of a blend, respectively. The mobility tensor and the energy function for each blend component are expressed in terms of these conformation tensors. The interaction effects are also included by energy expressions. The predictions of this family of “volume‐preserving” models are illustrated for a Hookean‐type energy function and several expressions of the modified mobility tensors. The results are illustrated for material functions in transient (start‐up and relaxation) and steady shear flows. The predictions are compared with experimental data taken from the literature for a miscible polymer blend. Study of the model sensitivity to its parameter shows that model predictions can cover a wide range of rheological behavior generally observed for multi‐component miscible polymer blends in steady and transient shear flows.Experimental data and model predictions for steady shear viscosity for HDPE/LDPE blends.imageExperimental data and model predictions for steady shear viscosity for HDPE/LDPE blends.

On the Duffin-Kemmer-Petiau formulation of the covariant Hamiltonian dynamics in field theory

We show that the De Donder-Weyl (DW) covariant Hamiltonian field equations of any field can be written in universal Duffin-Kemmer-Petiau (DKP) matrix form. As a consequence, the (modified) DKP matrices β μ (5 × 5 in four space-time dimensions) are of universal significance for all fields admitting the DW Hamiltonian formulation, not only for a scalar field, and can be viewed as field-theoretic analogues of the symplectic matrix, leading to the “ k-symplectic” ( k=4) structure. We also briefly discuss what could be viewed to be the covariant Poisson bracket given by β-matrices and the corresponding Poisson bracket formulation of DW Hamiltonian field equations.

On the Hamiltonian formulation of the quasi-hydrostatic equations

A variational approach to the Euler-Lagrange equations for the quasi-hydrostatic model is presented together with an explicit Hamiltonian (Poisson bracket) formulation. the bracket acts on functionals of the zonal and meridional angular momentum. Features of the relationships between the Hamiltonian formulations of the equations for an ideal fluid, the hydrostatic primitive equations and the quasi-hydrostatic equations are discussed.

On the Hamiltonian formulation of the quasi‐hydrostatic equations

AbstractA variational approach to the Euler‐Lagrange equations for the quasi‐hydrostatic model is presented together with an explicit Hamiltonian (Poisson bracket) formulation. the bracket acts on functionals of the zonal and meridional angular momentum. Features of the relationships between the Hamiltonian formulations of the equations for an ideal fluid, the hydrostatic primitive equations and the quasi‐hydrostatic equations are discussed.

Symmetry and integrable canonical flows

Following the method of a group theoretic formulation of rigid body dynamics, we construct an elementary proof that f commuting generators of symmetries of an f degree-of-freedom Hamiltonian system yield integrability of the dynamics in the form of f independent translations in phase space. The integrability of the dynamical system follows directly from the trivial integrability of a particular set of group parameter velocities that are nonintegrable in the absence of symmetry, and does not rely at all upon any assumption of separability of the Hamiltonian-Jacobi partial differential equation. Our method relies upon Hamel's explanation of when one can and cannot choose group parameters as generalized coordinates, and uses the Poisson bracket formulation of mechanics that is familiar to physicists. We formally extend Euler's theorem on rigid body motions to other transformation groups for Hamiltonian flows in phase space, and also note the analogy between nonholonomic coordinates in classical mechanics and uncertainty principles in quantum mechanics.

An algebraic approach for solving mechanical problems

An algebraic approach is used to obtain the time evolution of physical systems based on the Poisson bracket formulation of classical mechanics. This method is applied in many examples, including three-dimensional problems.

Poisson bracket formulation of viscoelastic flow equations of differential type: A unified approach

The Hamiltonian formulation of equations in continuum mechanics through a generalized bracket operation is shown here to reproduce a variety of incompressible viscoelastic fluid models, including the Giesekus model (with particular cases the upper‐convected Maxwell and the Oldroyd‐B models), the FENE–P dumbbell, the Phan‐Thien/Tanner, the Leonov, the Bird/DeAguiar, and the bead–spring chain models. The analysis allows comparison of the differential models on a more fundamental level than previously possible by reformulating the equations in terms of the Hamiltonian (system energy) and the dissipation of the system expressed as functionals involving the velocity vector and structural parameter(s). In fact, all of these models involve only slight variations of the same general Hamiltonian and the dissipation tensor. An advantage of this formulation is the establishment of thermodynamic admissibility criteria which in complex flows can shed light on the range of validity and/or faithfulness of the numerical calculations involving the above models. The usefulness of the generalized bracket formulation lies in the systematic approach that it provides in addressing one of the fundamental problems that the engineer working with complex materials has to deal with: how to transfer information that has been painstakingly provided by the physical chemist, addressing fundamental problems on a molecular level, from the microscopic scale to the macroscopic level where the engineer actually needs the model in dealing with everyday industrial problems. It is hoped that this new formulation can be used in the future to systematically generate continuum constitutive models, which are thermodynamically consistent, and based on microscopic analysis. Thus, it is the purpose here to narrow the gap between detailed (molecular) microscopic descriptions of the motions of polymer chains and (macroscopic phenomenological) continuum approaches. We believe that the generalized bracket formulation, due to its inherent simplicity and symmetry, has the potential to provide an answer to very complex situations, such as multicomponent structured media and coupled transport phenomena.

Poisson bracket formulation of incompressible flow equations in continuum mechanics

The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets is presented for a number of incompressible fluids, including the Euler inviscid fluid, the Newtonian viscous fluid, a perfectly elastic Poisson brackets is presented for a number of incompressible fluids, including the Euler inviscid fluid, the Newtonian viscous fluid, a perfectly elastic medium, the upper‐convected Maxwell, and the Oldroyd‐B viscoelastic fluids. The analysis, expanding previous results reported by Grmela, leads to a generalized Poisson bracket formalism from which all of the above‐mentioned cases can be recovered. Furthermore, the Poisson bracket formulation can easily incorporate model changes, as shown in the application of the hydrodynamic interaction correction to the Hookean dumbbell (upper‐convected Maxwell) model. The Hamiltonian formulation is fully explained here through a novel interpretation of the functional derivative through which the constraints of the flow are incorporated into the continuum equations. The Poisson bracket formulation, as developed here, can be used for the systematic development of constitutive equations of material behavior. This new approach allows a significant reduction in the number of arbitrary parameters and assumptions, compared to the traditional continuum modeling procedures. Simultaneously, the Hamiltonian formulation provides more structure to the system of equations and an easier interpretation of the various terms present in the constitutive equations.

Generalized constitutive equation for polymeric liquid crystals Part 1. Model formulation using the Hamiltonian (poisson bracket) formulation

The Hamiltonian formulation of equations in continuum mechanics through Poisson brackets, developed in Z.R. Iwinski and L.A. Turski, Lett. Appl. Eng. Sci., 4 (1976), 179–191, P.J. Morrison and J.M. Greene, Phys. Rev. Lett., 45 (1980) 790–794, I.E. Dzyaloshinskii and G.E. Volovick, Ann. Phys., 125 (1980) 67–97, D.D. Holm, J.E. Marsden, T. Ratiu and A. Weinstein, Phys. Rep., 123 (1985) 1–116, M. Grmela, Phys. Lett. A, 130 (1988) 81–86, and A.N. Beris and B.J. Edwards, J. Rheol., 34 (1990) 55–78, for a class of incompressible fluids, is used here in order to generate a constitutive equation for the stress and the order parametr tensor for a polymeric liquid crystal. A free energy expression, of the type used by Doi in his theory for concentrated solutions of rigid rods, is used in addition to the Frank elasticity expression employed in the Leslie—Ericksen—Parodi (LEP) theory to model the effect of spatial gradients in the liquid crystalline microstructure. For homogeneous systems, the analysis leads to a model which is equivalent to a generalization of Doi theory out to fourth-order terms in S . Truncating this model at second-order terms gives the Doi equations exactly. To evaluate the expanded model, results for steady simple shear and extensional flows are compared against the Doi model predictions. The constitutive equation resulting from the expanded model is compared against the LEP constitutive equation and the parameters between the two are correlated. The additional stress terms for non-homogeneous systems reduce to a recently presented (B.J. Edwards and A.N. Beris, J. Rheol., 33 (1989) 1189–1193; M. Grmela, Phys. Lett. A, 137 (1989) 342–348) generalization of the Ericksen stress expression in terms of the second-order parameter tensor. The model presented is a generalization and extension of the order-parameter-based theory of Doi which allows a greater flexibility in describing the rheological properties of polymeric liquid crystalline systems