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We study a holomorphic Poisson structure defined on the linear space $S(n,d):= {\rm Mat}_{n\times d}(\mathbb{C}) \times {\rm Mat}_{d\times n}(\mathbb{C})$ that is covariant under the natural left actions of the standard ${\rm GL}(n,\mathbb{C})$ and ${\rm GL}(d,\mathbb{C})$ Poisson-Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the $d=1$ special case gives a Poisson structure on $S(n,1)$, and we construct a local Poisson map from the Cartesian product of $d$ independent copies of $S(n,1)$ into $S(n,d)$, which is a holomorphic diffeomorphism in a neighborhood of zero. The Poisson structure on $S(n,d)$ is the complexification of a real Poisson structure on ${\rm Mat}_{n\times d}(\mathbb{C})$ constructed by the authors and Marshall, where a similar decoupling into $d$ independent copies was observed. We also relate our construction to a Poisson structure on $S(n,d)$ defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars-Schneider system by Hamiltonian reduction.

Developing the idea of increasing the number of structural elements in the unit cell of a quasi-one-dimensional lattice as applied to the semi-discrete integrable systems of nonlinear Schrodinger type, we construct the zero-curvature representation for the general integrable nonlinear system on a lattice with three structural elements in the unit cell. The integrability of the obtained general system permits to find explicitly a number of local conservation laws responsible for the main features of system dynamics and in particular for the so-called natural constraints separating the field variables into the basic and the concomitant ones. Thus, considering the reduction to the semi-discrete integrable system of nonlinear Schrodinger type, we revealed the essentially nontrivial impact of concomitant fields on the Poisson structure and on the whole Hamiltonian formulation of system dynamics caused by the nonzero background values of these fields. On the other hand, the zero-curvature representation of a ge...

In this note we present a simple formula for the Bruhat Poisson structure on complex projective spaces in terms of the momentum coordinates. We also give a simple description of a family of functions in involution on compact Hermitian symmetric spaces obtained via the bi-Hamiltonian approach using the Bruhat Poisson structure and an invariant symplectic structure. We compute these functions explicitly on CPn and relate them to the Gelfand–Tsetlin coordinates. We also show how the Lenard scheme can be applied.

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie–Poisson bracket on Se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Foppl equilibrium solutions using the energy-Casimir method.

A Poisson realization of the simple real Lie algebra so*(4n) on the phase space of each Sp(1)-Kepler problem is exhibited. As a consequence, one obtains the Laplace-Runge-Lenz vector for each classical Sp(1)-Kepler problem. The verification of these Poisson realizations is greatly simplified via an idea of Weinstein. The totality of these Poisson realizations is shown to be equivalent to the canonical Poisson realization of so*(4n) on the Poisson manifold T*H*n/Sp(1). (Here H*n≔Hn\{0} and the Hamiltonian action of Sp(1) on T*H*n is induced from the natural right action of Sp(1) on H*n.)

We construct two Poisson structures πTM and πTM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (TM,πTM,λ) from a bi-Hamiltonian manifold (M,π1,π2). This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πTM, πTM,λ, πTg*, and πTg*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.

A 3×3 discrete eigenvalue problem and corresponding discrete soliton equations are proposed. Under a constraint between the potentials and eigenfunctions, the 3×3 discrete eigenvalue problem is nonlinearized into an integrable Poisson map with a Lie–Poisson structure. Further, a reduction of the Lie–Poisson structure on the co-adjoint orbit yields the standard symplectic structure. The Poisson map is reduced to the usual symplectic map when it is restricted on the leaves of the symplectic foliation.

13 págs., A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively. © 1997 American Institute of Physics., This work has been partially supported through grants DGICYT (Spain) (Projects Nos. PB94-0106 and PB94-0633-C02-02), Consejería de Educación del Gobierno de Canarias, and UNED (Spain).

There is a family of degenerate AKNS systems for which the full theory of generic AKNS systems does not directly extend. The linear space of potentials still has a natural Poisson structure, but the scattering method used by Beals and Sattinger to show complete integrability for the generic AKNS systems fails for the degenerate case. A Poisson structure is not induced on the scattering side as in the generic case. As a consequence, the problem of complete integrability for degenerate AKNS systems still is an open question. In addition, contrary to the generic case, the Lax pair gives flows for degenerate integrable systems that are nonlocal. In general, they do not exist, and they are no longer linear on the scattering side. Necessary conditions for their existence and for linear evolution in the scattering side are found.

In this paper, a class of degenerate (i.e., associated to a degenerate Poisson structure) completely integrable systems is studied which generalizes the so‐called odd and even master systems introduced and studied by Mumford and by Vanhaecke. It is shown that all these completely integrable systems, called the generalized master systems, admit a multi‐Hamiltonian formulation, and a systematic construction of this multi‐Hamiltonian structure is described.

In this paper, we study the Vlasov-Maxwell equations based on the Morrison-Marsden-Weinstein bracket. We develop Hamiltonian particle-in-cell methods for this system by employing finite element methods in space and splitting methods in time. In order to derive the semi-discrete system that possesses a discrete non-canonical Poisson structure, we present a criterion for choosing the appropriate finite element spaces. It is confirmed that some conforming elements, e.g., Nedelec's mixed elements, satisfy this requirement. When the Hamiltonian splitting method is used to discretize this semi-discrete system in time, the resulting algorithm is explicit and preserves the discrete Poisson structure. The structure-preserving nature of the algorithm ensures accuracy and fidelity of the numerical simulations over long time.

Under what algebraic conditions n‐dimensional Lotka–Volterra equations are Hamiltonian for a suitable Poisson structure is investigated herein.

We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy contains both the higher KdV hierarchy and multi--field representation of KP hierarchy as sub--systems and naturally appears in multi--matrix models. The N+2M-1 coordinates or fields of this hierarchy satisfy two algebras of compatible Poisson brackets which are {\it local} and {\it polynomial}. Each Poisson structure generate an extended W_{1+\infty} and W_\infty algebra, respectively. We call W(N, M) the generating algebra of the extended W_\infty algebra. This algebra, which corresponds with the second Poisson structure, shares many features of the usual $W_N$ algebra. We show that there exist M distinct reductions of the (N, M)--th KdV hierarchy, which are obtained by imposing suitable second class constraints. The most drastic reduction corresponds to the (N+M)--th KdV hierarchy. Correspondingly the W(N, M) algebra is reduced to the W_{N+M} algebra. We study in detail the dispersionless limit of this hierarchy and the relevant reductions.

Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.

It is shown that the Poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one‐form in three dimensions. Advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability. Techniques for finding Poisson structures are presented and applied to various examples such as the Halphen system which has been studied as the two‐monopole problem by Atiyah and Hitchin. It is shown that the Halphen system can be formulated in terms of a flat SL(2,R)‐valued connection and belongs to a nontrivial Godbillon–Vey class. On the other hand, for the Euler top and a special case of three‐species Lotka–Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi‐Hamiltonian structures. The globally integrable bi‐Hamiltonian case is a linear and the SL(2,R) structure is a quadratic unfolding of an integrable one‐form in 3+1 dimensions. It is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of Poisson structure and some new techniques for incorporating arbitrary constants into the Poisson one‐form are presented herein. This leads to some extensions, analogous to q extensions, of Poisson structure. The Kermack–McKendrick model and some of its generalizations describing the spread of epidemics, as well as the integrable cases of the Lorenz, Lotka–Volterra, May–Leonard, and Maxwell–Bloch systems admit globally integrable bi‐Hamiltonian structure.

A universal functional Ito formula, recently discovered within the *‐Ito noncommutative algebraic approach, is described herein. The formula unifies the classical Ito formulas for Wiener, Poisson, and quantum cases, extends the chain rule df(x1,...,xl) = ∑f’i(x)dxi to the case of noncommuting stochastically differentiable operators Xi(t), and also admits a nonadapted generalization.

The relation between the R‐matrix approach to the construction of bi‐Hamiltonian structures and the reduction of Poisson manifolds is discussed. The crucial role of the reduction process is emphasized by showing, on the basis of specific examples, that the knowledge of a R matrix may not be sufficient. The reduction of the Poisson structures related with R from the entire algebra to the Lax submanifold may become an obstruction in constructing the bi‐Hamiltonian structures of a specific hierarchy of integrable equations.

There are two known integrable N=2 space supersymmetric extensions of the KdV equation. Both can be written as Hamiltonian systems with a common Poisson structure which corresponds to the N=2 supersymmetric form of the second KdV structure. By analyzing the conservation laws of the one parameter family of equations that share this Hamiltonian structure, one finds that a third system is singled out by the existence of nontrivial conservation laws. It is conjectured to be integrable.

Following the considerable progress in nanoribbon technology, we propose to model the nonlinear Frenkel-like excitations on a triangular-lattice ribbon by the integrable nonlinear ladder system with the background-controlled intersite resonant coupling. The system of interest arises as a proper reduction of first general semidiscrete integrable system from an infinite hierarchy. The most significant local conservation laws related to the first general integrable system are found explicitly in the framework of generalized recursive approach. The obtained general local densities are equally applicable to any general semidiscrete integrable system from the respective infinite hierarchy. Using the recovered second densities, the Hamiltonian formulation of integrable nonlinear ladder system with background-controlled intersite resonant coupling is presented. In doing so, the relevant Poisson structure turns out to be essentially nontrivial. The Darboux transformation scheme as applied to the first general semidiscrete system is developed and the key role of Backlund transformation in justification of its self-consistency is pointed out. The spectral properties of Darboux matrix allow to restore the whole Darboux matrix thus ensuring generation one more soliton as compared with a priori known seed solution of integrable nonlinear system. The power of Darboux-dressing method is explicitly demonstrated in generating the multicomponent one-soliton solution to the integrable nonlinear ladder system with background-controlled intersite resonant coupling.

The damped harmonic oscillator is one of the most studied systems with respect to the problem of quantizing dissipative systems. Recently Chandrasekar et al. [J. Math. Phys. 48, 032701 (2007)] applied the Prelle–Singer method to construct conserved quantities and an explicit time-independent Lagrangian and Hamiltonian structure for the damped oscillator. Here we describe the associated Poisson bracket which generates the continuous flow, pointing out that there is a subtle problem of definition on the whole phase space. The action-angle variables for the system are also presented, and we further explain how to extend these considerations to the discrete setting. Some implications for the quantum case are briefly mentioned.

Each of the local isometry groups arising in three-dimensional (3d) gravity can be viewed as a group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for the case of Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson–Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, among others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,R). We also compute the Poisson structure on the dual Poisson–Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.

Finite-dimensional reductions of the 2D dispersionless Toda hierarchy, constrained by the ``string equation'' are studied. These include solutions determined by polynomial, rational or logarithmic functions, which are of interest in relation to the ``Laplacian growth'' problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchies is also derived

Two cohomology classes associated to groups of transformations (symplectic or not) of Hamiltonian and Lagrangian systems are studied. A geometrical interpretation of the family of cocycles arising from a class of nonsymplectic actions is given in terms of the Poisson structure of the phase space of the system. These ideas are used to study nongauge (i.e., anomalous) groups of transformations of (locally or globally defined) Lagrangian systems. In particular, well‐known results about the magnetic monopole system are described in this context and some hints relating Yang–Mills anomalies with nonsymplectic groups of transformations are given.

The equations of the two‐ (or multi‐) fluid model of plasma physics are recast in Hamiltonian form, following general methods of symplectic geometry. The dynamical variables are the fields of physical interest, but are noncanonical, so that the Poisson bracket in the theory is not the standard one. However, it is a skew‐symmetric bilinear form which, from the method of derivation, automatically satisfies the Jacobi identity; therefore, this noncanonical structure has all the essential properties of a canonical Poisson bracket.

A local normal form under canonical transformation is found for n independent functions of 2n variables, with the condition that the Poisson bracket of each pair of the functions be constant. The normal form, closely related to work of Lie, is used to prove a conjecture of Avez on 1‐forms in involution and to obtain a criterion for n independent functions of 2n variables to be extendable to a canonical coordinate system. The last result has been obtained in different ways by Lie and Kruskal.

It is shown that the usual moment map J: T*(R3−{0})]so*(2,4) of the Kepler problem can be generalized to include the magnetic term of the Dirac monopole.

The canonical Poisson structure of the microscopic Lagrangian is used to deduce the noncanonical Poisson structure for the macroscopic Hamiltonian dynamics of a compressible neutral fluid and of fluid electrodynamics.