Lie-Poisson Structure Research Articles

Stabilization of an underactuated bottom‐heavy airship via interconnection and damping assignment

AbstractThis paper focuses on feedback stabilization of a neutrally buoyant and bottom‐heavy airship actuated by only five independent controls (with the rolling motion underactuated). The airship is modelled as an eudipleural submerged rigid body whose dynamics is formulated as a Hamiltonian system with respect to a Lie–Poisson structure. By exploiting the geometrical structure and using the so‐called interconnection and damping assignment (IDA) passivity‐based methodology for port‐controlled Hamiltonian systems, state feedback control laws asymptotically stabilizing two typical motions are designed via La Salle invariance principle and Chetaev instability theorem. Simulation results verify the control laws. Copyright © 2007 John Wiley & Sons, Ltd.

Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures, called affine or modified Lie-Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds

The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

Super Poisson–Lie structure on [formula omitted] via [formula omitted

This paper concerns a super Poisson–Lie structure on the real Lie supergroup S U ( m | n ) . In fact, it turns out that the realification of the complex Lie supergroup S L ( m | n , C ) is a double of S U ( m | n ) i.e. it is endowed with a structure of super Poisson–Lie which brings down on the supergroup S U ( m | n ) . We show that the dual Poisson–Lie supergroup of S U ( m | n ) is s ( A N ) . Reciprocally, s ( A N ) inherits a super Poisson–Lie structure from the realification of S L ( m | n , C ) such that its dual Poisson–Lie supergroup is S U ( m | n ) .

Examples of quantisation of Poisson manifolds

In quantum physics, the operators associated with the position and the momentum of a article are unbounded operators and C ∗ -algebraic quantisation does therefore not deal with such operators. In the present article, I propose a quantisation of the Lie–Poisson structure of the dual of a Lie algebroid which deals with a big enough class of functions to include the above-mentioned example. As an application, I show with an example how the quantisation of the dual of the Lie algebroid associated to a Poisson manifold can lead to a quantisation of the Poisson manifold itself. The example, I consider is the torus with constant Poisson structure, in which case I recover its usual C ∗ -algebraic quantisation.

Integrability of Invariant Metrics on the Diffeomorphism Group of the Circle

Each Hk Sobolev inner product (k ≥ 0) defines a Hamiltonian vector field Xk on the regular dual of the Lie algebra of the diffeomorphism group of the circle. We show that only X0 and X1 are bi-Hamiltonian relative to a modified Lie-Poisson structure.

Integrability of invariant metrics on the Virasoro group

Each right-invariant metric on the Virasoro group induces a Hamiltonian vector field on the dual of the Lie algebra vir equipped with the canonical Lie–Poisson structure. We show that the Hamiltonian vector fields X k induced by the metrics given at the identity by the H k Sobolev inner products, k ⩾ 0 , are bi-Hamiltonian relative to a modified Lie–Poisson structure only for k = 0 and k = 1 .

Efficient Lie-Poisson Integrator for Secular Spin Dynamics of Rigid Bodies

A fast and efficient numerical integration algorithm is presented for the problem of the secular evolution of the spin axis. Under the assumption that a celestial body rotates around its maximum moment of inertia, the equations of motion are reduced to the Hamiltonian form with a Lie-Poisson bracket. The integration method is based on the splitting of the Hamiltonian function, and so it conserves the Lie-Poisson structure. Two alternative partitions of the Hamiltonian are investigated, and second-order leapfrog integrators are provided for both cases. Non-Hamiltonian torques can be incorporated into the integrators with a combination of Euler and Lie-Euler approximations. Numerical tests of the methods confirm their useful properties of short computation time and reliability on long integration intervals.

Two (2+1)-dimensional soliton equations and their quasi-periodic solutions

Two (2 + 1)-dimensional soliton equations and their decomposition into the mixed (1 + 1)-dimensional soliton equations are proposed. With the help of nonlinearization approach, the Lenard spectral problem related to the mixed soliton hierarchy is turned into a completely integrable Hamiltonian system with a Lie–Poisson structure on the Poisson manifold R 3 N . The Abel–Jacobi coordinates are introduced to straighten out the Hamiltonian flows. Based on the decomposition and the theory of algebra curve, the explicit quasi-periodic solutions for the (1 + 1)- and (2 + 1)-dimensional soliton equations are obtained.

The Lie-Poisson structure of the Euler equations of an ideal fluid

This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R^n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C^1 from the Sobolev class H^s to itself (where s > (n=2) + 1). The idea of how this diculty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure.

Poisson sigma model over group manifolds

We study the Poisson sigma model which can be viewed as a topological string theory. Mainly we concentrate our attention on the Poisson sigma model over a group manifold G with a Poisson–Lie structure. In this case the flat connection conditions arise naturally. The boundary conditions (D-branes) are studied in this model. It turns out that the D-branes are labelled by the coisotropic subgroups of G . We give a description of the moduli space of classical solutions over Riemann surfaces both without and with boundaries. Finally we comment briefly on the duality properties of the model.

Moyal Deformation of KdV and Virasoro Action

The Lie algebra of pseudodifferential symbols Ψ D ( S 1 ) on S 1 has a natural Lie–Poisson structure. The Moyal deformation of dispersionless Korteweg–de Vries (KdV) equation is constructed from th...

Integrability of Characteristic Hamiltonian Systems on Simple Lie Groups with Standard Poisson Lie Structure

Phase space of a characteristic Hamiltonian system is a symplectic leaf of a factorizable Poisson Lie group. Its Hamiltonian is a restriction to the symplectic leaf of a function on the group which is invariant with respect to conjugations. It is shown in this paper that such a system is always integrable.

Quantization of Lie$ndash$Poisson structures by peripheric chains

The quantization properties of composite peripheric twists are studied. Peripheric chains of extended twists are constructed for U(sl(N)) in order to obtain composite twists with sufficiently large carrier subalgebras. It is proved that the peripheric chains can be enlarged with additional Reshetikhin and Jordanian factors. This provides the possibility to construct new solutions to Drinfeld equations and, thus, to quantize new sets of Lie-Poisson structures. When the Jordanian additional factors are used the carrier algebras of the enlarged peripheric chains are transformed into algebras of motion of the form G_{JB}^{P}={G}_{H}\vdash {G}_{P}. The factor algebra G_{H} is a direct sum of Borel and contracted Borel subalgebras of lower dimensions. The corresponding omega--form is a coboundary. The enlarged peripheric chains F_{JB}^{P} represent the twists that contain operators external with respect to the Lie-Poisson structure. The properties of new twists are illustrated by quantizing r-matrices for the algebras U(sl(3)), U(sl(4)) and U(sl(7)).

An integrable Poisson map generated from the eigenvalue problem of the Lotka-Volterra hierarchy

A 3?3 discrete eigenvalue problem associated with the Lotka-Volterra hierarchy is studied and the corresponding nonlinearized one, an integrable Poisson map with a Lie-Poisson structure, is also presented. Moreover, a 2?2 nonlinearized eigenvalue problem, which also begets the Lotka-Volterra hierarchy, is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation.

Boundary G/G theory and topological Poisson-Lie sigma model

We study a boundary version of the gauged WZW model with a Poisson–Lie group G as the target. The Poisson–Lie structure of G is used to define the Wess–Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson–Lie group G* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov–Tian–Shansky.

On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups

We provide an explicit description of symplectic leaves of a simply connected, connected, semisimple, complex Lie group equipped with the standard Poisson-Lie structure. This sharpens previously known descriptions of the symplectic leaves as connected components of certain varieties. Our main tool is the machinery of twisted generalized minors. They also allow us to present several quasi-commuting coordinate systems on every symplectic leaf. As a consequence, we construct new completely integrable systems on some special symplectic leaves.

Complex form, reduction and Lie–Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy

In this paper, the relation between the different restricted systems associated with the Heisenberg magnetic (HM) equation is studied by the reduction procedure. With the help of a Lie group homomorphism of SU (2) into SO (3) , the Euler–Rodriguez-type parameters are introduced to generate new finite-dimensional integrable system. It has shown that the resulting system, which is the nonlinearized spectral problem of HM hierarchy on C 2 N , is a Hamiltonian system in complex form. Further, Poisson reduction and Lie–Poisson structure are derived by the method of invariants. The reduced system is found to be a Hamiltonian system on the orbit space C 2 N / T N ≃ R 3 N , coinciding with the nonlinearized Lenard spectral problem. Moreover, the fully reduced systems on the leaves of the symplectic foliation are also given. Specifically, the reduction extended to the common level set of the complex cones is the usual 2×2 nonlinearized spectral problem. Finally, the integrability of the system with Lie–Poisson structure is proven by making use of the SO (3) symmetry.

POISSON STRUCTURES ON THE GALILEI GROUP

Poisson–Lie structures on the Galilei group are completely classified.

The Lie–Poisson representation of the nonlinearized eigenvalue problem of the Kac–van Moerbeke hierarchy

The Kac–van Moerbeke hierarchy is studied by a 3×3 discrete eigenvalue problem and the corresponding nonlinearized one an integrable Poisson map with a Lie–Poisson structure is also presented. Moreover, the 2×2 nonlinearized eigenvalue problem associated with the Kac–van Moerbeke hierarchy is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation.