Lie-Poisson Bracket Research Articles

Zhukovsky-Volterra top and quantisation ideals

In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics—the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two linear Poisson brackets. Using the quantisation ideal method, we have identified two distinct quantisations of the Zhukovsky-Volterra top. The first type corresponds to the universal enveloping algebras of so(3), leading to Lie-Poisson brackets in the classical limit. The second type can be regarded as a quantisation of the four-parametric inhomogeneous quadratic Poisson pencil. We discuss the relationships between the quantisations obtained in our paper, Sklyanin's quantisation of the Euler top, and Levin-Olshanetsky-Zotov's quantisation of the Zhukovsky-Volterra top.

Hamiltonian formalism for Bose excitations in a plasma with a non-Abelian interaction I: Plasmon – hard particle scattering

Hamiltonian theory for collective longitudinally polarized gluon excitations (plasmons) interacting with classical high-energy test color-charged particle propagating through a high-temperature gluon plasma is developed. A generalization of the Lie-Poisson bracket to the case of a continuous medium involving bosonic normal field variable ak⁎a and a non-Abelian color charge Qa is performed and the corresponding Hamilton equations are presented. The canonical transformations including simultaneously both bosonic degrees of freedom of the soft collective excitations and degree of freedom of hard test particle connecting with its color charge in the hot gluon plasma are written out. A complete system of the canonicity conditions for these transformations is derived. The notion of the plasmon number density Nkaa1′, which is a nontrivial matrix in the color space, is introduced. An explicit form of the effective fourth-order Hamiltonian describing the elastic scattering of a plasmon off a hard color particle is found and the self-consistent system of Boltzmann-type kinetic equations taking into account the time evolution of the mean value of the color charge of the hard particle is obtained. On the basis of these equations, a model problem of the interaction of two infinitely narrow wave packets is considered. A system of nonlinear first-order ordinary differential equations defining the dynamics of the interaction of the colorless Nkl and color Wkl components of the plasmon number density is derived. The problem of determining the third- and fourth-order coefficient functions entering into the canonical transformations of the original bosonic variable ak⁎a and color charge Qa is discussed. With the help of the coefficient functions obtained, a complete effective amplitude of the elastic scattering of plasmon off hard test particle is written out.

Nambu Bracket for 3D Ideal Fluid Dynamics and Magnetohydrodynamics

Abstract The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by a Hamilton equation with respect to the Lie–Poisson bracket. The Nambu bracket manifestly represents the Lie–Poisson structure in terms of derivatives of the Casimir invariants. We construct a compact Nambu bracket representation for the 3D ideal MHD equations with the use of three Casimirs for the second Hamiltonians, the total entropy, and the magnetic and cross-helicities, whose coefficients are all constant. The Lie–Poisson bracket induced by this Nambu bracket does not coincide with the original one, but is supplemented by terms with an auxiliary variable. The supplemented Lie–Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether’s theorem, we show that the cross-helicity is an integral invariant associated with the particle-relabeling symmetry. Employing a Lagrange label function as the independent variable in the variational framework facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground as Noether’s theorem.

On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described.

Polynomial algebras from Lie algebra reduction chains [formula omitted] [formula omitted][formula omitted][formula omitted

We reexamine different examples of reduction chains g⊃g′ of Lie algebras in order to show how the polynomials determining the commutant with respect to the subalgebra g′ leads to polynomial deformations of Lie algebras. These polynomial algebras have already been observed in various contexts, such as in the framework of superintegrable systems. Two relevant chains extensively studied in Nuclear Physics, namely the Elliott chain su(3)⊃so(3) and the chain so(5)⊃su(2)×u(1) related to the Seniority model, are analysed in detail from this perspective. We show that these two chains both lead to three-generator cubic polynomial algebras, a result that paves the way for a more systematic investigation of nuclear models in relation to polynomial structures arising from reduction chains. In order to show that the procedure is not restricted to semisimple algebras, we also study the chain Sˆ(3)⊃sl(2,R)×so(2) involving the centrally-extended Schrödinger algebra in (3+1)-dimensional space–time. The approach chosen to construct these polynomial algebras is based on the use of the Lie-Poisson bracket, so that all the results are presented in the Poisson (commutative) setting. The advantage of considering this approach vs the enveloping algebras is emphasized, commenting on the main formal differences between the polynomial Poisson algebras and their noncommutative analogue. As an illustrative example of the latter, the three-generator cubic algebra associated to the Elliott chain is reformulated in the Lie algebraic (noncommutative) setting.

SU(2) Lie-Poisson algebra and its descendants

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R3. The starting point of our derivation is the discrete Frenet frame assigned at each vertix of the string. Then the link vector that connect the neighbouring vertices assigns the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighbouring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial and novel Poisson structure that engages four neighbouring vertices.

Local Conservation Laws in a Nonlinear Electrodynamics

By using a generalization of the Lie–Poisson brackets for the dual Maxwell and Born–Infeld field strength tensors, we construct the gauge invariant axial-vector conserved currents for Born–Infeld and Heisenberg–Euler nonlinear electrodynamics in the 4-dimensional Minkowski space-time. The infinite hierarchies of the currents given by Lie brackets for generally covariant conserved vector and axial vector currents are established. These currents are conserved upon action of the gravitational fields, but the conservation is broken in the Einstein–Cartan theory (over a Riemann–Cartan space-time). The axial-vector currents are conserved only in the (3 + 1)-dimensional space-time.

Double bracket formulation for the distribution function approach to multibead-chain suspensions

A suspension of elastic chains of small beads in a Newtonian fluid is a common model for a viscoelastic polymer solution. The configuration of these multibead chains is described by a distribution function that evolves according to a Liouville or Fokker–Planck equation. The evolution of these multibead-chain suspensions is described using a double bracket formulation with a Hamiltonian functional. The conservative part of the dynamics is described by a Poisson bracket, and the dissipative part by an additional symmetric bracket. We treat the configuration space of multibead chains as a higher order tangent bundle. Lifting the fluid velocity field to the bundle leads naturally to a semidirect product Lie–Poisson bracket for the conservative dynamics. The elastic stress exerted by the multibead chains on the fluid then follows directly from the same Hamiltonian functional that governs the internal dissipative mechanics of the multibead chains. For chains with three or more beads, the possible bending of the chain introduces an angular momentum flux that is absent for chains with two beads. This flux appear as an asymmetric elastic stress whose antisymmetric part is the divergence of a rank-3 tensor, as in the Cosserats’ theory of couple stresses in media with no internal angular momentum density. We investigate the possibility of an exact closure, passing from a distribution function description to a closed internal state variable description of the fluid suspension, and obtain some sufficient conditions for their existence. The resulting exactly closable models are generalisations of the upper-convected Maxwell model to Hookean bead–spring chains instead of Hookean bead–spring pairs.

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

Provenance of classical Hamiltonian time crystals

Classical Hamiltonian systems with conserved charges and those with constraints often describe dynamics on a pre-symplectic manifold. Here we show that a pre-symplectic manifold is also the proper stage to describe autonomous energy conserving Hamiltonian time crystals. We explain how the occurrence of a time crystal relates to the wider concept of spontaneously broken symmetries; in the case of a time crystal, the symmetry breaking takes place in a dynamical context. We then analyze in detail two examples of timecrystalline Hamiltonian dynamics. The first example is a piecewise linear closed string, with dynamics determined by a Lie-Poisson bracket and Hamiltonian that relates to membrane stability. We explain how the Lie-Poisson brackets descents to a time-crystalline pre-symplectic bracket, and we show that the Hamiltonian dynamics supports two phases; in one phase we have a time crystal and in the other phase time crystals are absent. The second example is a discrete one dimensional model of a Hamiltonian chain. It is obtained by a reduction from the Q-ball Lagrangian that describes time dependent nontopological solitons. We show that a time crystal appears as a minimum energy domain wall configuration, along the chain.

Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

Abstract The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.

Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics.

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.

Complete families of commuting functions for coisotropic Hamiltonian actions

Let G be an algebraic group defined over a field F of characteristic zero with g=LieG. The dual space g⁎ equipped with the Lie–Poisson bracket is a Poisson variety and each irreducible G-invariant subvariety X⊂g⁎ carries the induced Poisson structure. We prove that there is a set {f1,...,fl}⊂F[X] of algebraically independent polynomial functions, which pairwise commute with respect to the Poisson bracket, such that l=(dim⁡X+tr.degF(X)G)/2. We also discuss several applications of this result to complete integrability of Hamiltonian systems on symplectic Hamiltonian G-varieties of corank zero and 2.

Asymptotic symmetries of three dimensional gravity and the membrane paradigm

The asymptotic symmetry group of three-dimensional (anti) de Sitter space with Brown-Henneaux boundary conditions is the two dimensional conformal group with central charge c = 3ℓ/2G. Usually the asymptotic charge algebra is derived using the symplectic structure of the bulk Einstein equations. Here, we derive the asymptotic charge algebra by a different route. First, we formulate the dynamics of the boundary as a 1+1-dimensional dynamical system. Then we realize the boundary equations of motion as a Hamiltonian system on the dual Lie algebra, {mathfrak{g}}^{ast } , of the two-dimensional conformal group. Finally, we use the Lie-Poisson bracket on {mathfrak{g}}^{ast } to compute the asymptotic charge algebra. This streamlines the derivation of the asymptotic charge algebra because the Lie-Poisson bracket on the boundary is significantly simpler than the symplectic structure derived from the bulk Einstein equations. It also clarifies the analogy between the infinite dimensional symmetries of gravity and fluid dynamics.

Dual pairs and regularization of Kummer shapes in resonances

We present an account of dual pairs and the Kummer shapes for $ n:m $ resonances that provides an alternative to Holm and Vizman's work. The advantages of our point of view are that the associated Poisson structure on $ \mathfrak{su}(2)^{*} $ is the standard $ (+) $-Lie-Poisson bracket independent of the values of $ (n,m) $ as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of $ (n,m) $. A similar result holds for $ n:-m $ resonance with a paraboloid and $ \mathfrak{su}(1,1)^{*} $. The result also has a straightforward generalization to multidimensional resonances as well.

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.

BMS3 invariant fluid dynamics at null infinity

We revisit the boundary dynamics of asymptotically flat, three dimensional gravity. The boundary is governed by a momentum conservation equation and an energy conservation equation, which we interpret as fluid equations, following the membrane paradigm. We reformulate the boundary’s equations of motion as Hamiltonian flow on the dual of an infinite-dimensional, semi-direct product Lie algebra equipped with a Lie–Poisson bracket. This gives the analogue for boundary fluid dynamics of the Marsden–Ratiu–Weinstein formulation of the compressible Euler equations on a manifold, M, as Hamiltonian flow on the dual of the Lie algebra of . The Lie group for boundary fluid dynamics turns out to be , with central charge c = 3/G. This gives a new derivation of the centrally extended, three-dimensional Bondi–van der Burg–Metzner–Sachs () group. The relationship with fluid dynamics helps to streamline and physically motivate the derivation. For example, the central charge, c = 3/G, is simply read off of a fluid equation in much the same way as one reads off a viscosity coefficient. The perspective presented here may useful for understanding the still mysterious four-dimensional BMS group.

Near-horizon BMS symmetries as fluid symmetries

The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity. Recently, Donnay et al. have derived an analogous symmetry group acting on black hole event horizons. For a certain choice of boundary conditions, it is a semidirect product of Diff(S2), the smooth diffeomorphisms of the twosphere, acting on C∞(S2), the smooth functions on the two-sphere. We observe that the same group appears in fluid dynamics as symmetries of the compressible Euler equations. We relate these two realizations of Diff(S2) ⋉ C∞(S2) using the black hole membrane paradigm. We show that the Lie-Poisson brackets of membrane paradigm fluid charges reproduce the near-horizon BMS algebra. The perspective presented here may be useful for understanding the BMS algebra at null infinity.

On the origin of dual Lax pairs and their r-matrix structure

We establish the algebraic origin of the following observations made previously by the authors and coworkers: (i) A given integrable PDE in $1+1$ dimensions within the Zakharov-Shabat scheme related to a Lax pair can be cast in two distinct, dual Hamiltonian formulations; (ii) Associated to each formulation is a Poisson bracket and a phase space (which are not compatible in the sense of Magri); (iii) Each matrix in the Lax pair satisfies a linear Poisson algebra a la Sklyanin characterized by the {\it same} classical $r$ matrix. We develop the general concept of dual Lax pairs and dual Hamiltonian formulation of an integrable field theory. We elucidate the origin of the common $r$-matrix structure by tracing it back to a single Lie-Poisson bracket on a suitable coadjoint orbit of the loop algebra ${\rm sl}(2,\CC) \otimes \CC (\lambda, \lambda^{-1})$. The results are illustrated with the examples of the nonlinear Schr\"odinger and Gerdjikov-Ivanov hierarchies.