Lie-Poisson Bracket Research Articles

Hamiltonian systems related to invariant Metrics

Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation which acts on the Lie algebra. By inducing the canonical symplectic structure of the coadjoint orbits to the orbits of the new representation, we study Hamiltonians on the orbits by determining the Hamiltonian vector fields for invariant functions and we find commuting conditions under the corresponding Lie-Poisson bracket.

A Hamiltonian electromagnetic gyrofluid model

An isothermal truncation of the electromagnetic gyrofluid model of Snyder and Hammett [Phys. Plasmas 8, 3199 (2001)] is shown to be Hamiltonian. The corresponding noncanonical Lie–Poisson bracket and its Casimir invariants are presented. The invariants are used to obtain a set of coupled Grad–Shafranov equations describing equilibria and propagating coherent structures.

Hamiltonian formulation and analysis of a collisionless fluid reconnection model

The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie–Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad–Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation is also used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.

Dual R-matrix integrability

Using the R-operator on a Lie algebra \(\mathfrak{g}\) satisfying the modified classical Yang-Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie-Poisson bracket on \(\mathfrak{g}^ * \). We consider examples of the Lie algebras \(\mathfrak{g}\) with the Kostant-Adler-Symes and triangular decompositions, their R-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which R-operators the constructed sets of functions also commute with respect to the R-bracket. We briefly discuss the Euler-Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals.

Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket

The dynamics of Vlasov kinetic moments is shown to be Lie–Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie–Poisson bracket is extended to anisotropic interactions.

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.

Classification of compatible Lie-Poisson brackets on the manifold e*(3)

A classification of compatible Lie-Poisson brackets on e*(3) is constructed. The corresponding bi-Hamiltonian systems are the classical integrable cases of the Euler-Poisson and Kirchhoff equations which describe the motion of a solid body. Bibliography: 12 titles.

Compatible Lie-Poisson brackets on the Lie algebras e(3) and so(4)

We completely classify the compatible Lie-Poisson brackets on the dual spaces of the Lie algebras e(3) and so(4). The corresponding bi-Hamiltonian systems are the spinning tops corresponding to the classical cases of integrability of the Euler equations, the Kirchhoff equations, and the Poincare-Zhukovskii equations.

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies M n for univalent functions. Based on the Löwner–Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then M n are defined as sub-Riemannian manifolds. Given a Lie–Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system to calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case M 3 .

Integrable quadratic Hamiltonians with a linear Lie-Poisson bracket

Quadratic Hamiltonians with a linear Lie-Poisson bracket have a number of applications in mechanics. For example, the Lie-Poisson bracket e(3) includes the Euler-Poinsot model describing motion of a rigid body around a fixed point under gravity and the Kirchhoff model describes the motion of a rigid body in ideal fluid. Among the applications with a Lie-Poisson bracket so(4) and so(3, 1) is the description of free rigid body motion in a space of constant curvature. Advances in computer algebra algorithms, in implementations and hardware, together allow the computation of Hamiltonians with higher degree first integrals providing new results in the classic search for integrable models. A computer algebra module enabling related computations in a 3-dimensional vector formalism is described.

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.

Euler-Poincaré Formalism of Coupled KdV Type Systems and Diffeomorphism Group on S1

This paper describes a wide class of coupled KdV equations. The first set of equations directly follow from the geodesic flows on the Bott-Virasoro group with a complex field. But the set of 2-component systems of nonlinear evolution equations, which includes dispersive water waves, Ito's equation, many other known and unknown equations, follow from the geodesic flows of the right invariant L 2 metric on the semidirect product group , where Diff( S 1 ) is the group of orientation preserving diffeomorphisms on a circle. We compute the Lie-Poisson brackets of the Antonowicz-Fordy system, and the mode expansion of these beackets yield the twisted Heisenberg-Virasoro algebra. We also give an outline to study geodesic flows of a H 1 metric on .

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.

The Lie-Poisson structure of the LAE-α equation

This paper shows that the time t map of the averaged Euler equa- tions, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding dieomorphism groups. It is also shown that the geodesic spray for Neumann and mixed boundary conditions is smooth, a result already known for Dirichlet boundary conditions.

Application of Poisson maps on coadjoint orbits of Sp(6,R) group to many body dynamics

The canonical transformation approach generated by the semisimple subgroup GCM(3)⊂Sp(6,R) is applied to reduction of the Lie–Poisson bracket on coadjoint orbits of the Sp(6,R) group and the Poisson coalgebra spO*(6) is determined. Investigating the construction of the N-particle phase, induced by this reduction, we identify the Poisson coalgebra spO*(6) as the algebra of quadratic O(K), K≡N−1 invariant forms on symplectic 6 K−12 dimensional phase space T*[O(K−3)\O(K)], K⩾3. The general classification scheme of Poisson orbits for spO*(6) is found and applied to the classification of coadjoint orbits of the Sp(6,R) group occurring in the decomposition of N-particle phase spaces. We show that the spO*(k), k=4,6 Poisson action on some class of surfaces determined by Casimir invariants is not transitive. The Poisson maps for all classes of orbits spO*(4) and spO*(6) are found. The quantum unitary irreducible representations of spO*(4) are obtained.

Geometrical theory of two-dimensional hydrodynamics with special reference to a system of point vortices

Two-dimensional hydrodynamics is formulated geometrically with a metric defined in terms of stream function and vorticity for flows of an inviscid incompressible fluid in unbounded space. Based on this, the vorticity equation is derived as a geodesic equation over a group of diffeomorphisms of a fluid at rest at infinity with vorticity having compact supports, which is also rewritten in the form of a Hamilton's equation with a Lie–Poisson bracket. In particular, this formulation is applied to a flow induced by a finite number ( N) of point vortices. It is shown that the geodesic equation reduces to the well-known Hamiltonian system of N point vortices. This is considered as a dynamics of an internal space of finite degrees of freedom, as against the former background dynamical system described by a group of diffeomorphisms of infinite dimensions and Riemannian geometry. It is shown that the dynamical trajectories of the system of N degrees of freedom governed by Hamiltonian system of equations are the projections of the geodesics of ( N+1)-dimensional Finslerian space associated with a fundamental function obtained from the original Lagrangian of N point vortices. The Finsler geometry provides us curvature tensors and equation of geodesic deviation. As an example, the Finslerian geometry is applied to a system of a vortex pair, in which sum of two vortex strengths vanishes and the pair makes steady translational motion with a constant velocity. The scalar curvature of the motion is found to be zero, and its relation to geodesic deviation is considered.

Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields

The Hamiltonian structure of the inhomogeneous layer models for geophysical fluid dynamics devised by Ripa [Geophys. Astrophys. Fluid Dyn. 70, 85 (1993)] involves the same Poisson bracket as a Hamiltonian formulation of shallow water magnetohydrodynamics in velocity, height, and magnetic flux function variables. This Poisson bracket becomes the Lie–Poisson bracket for a semidirect product Lie algebra under a change of variables, giving a simple and direct proof of the Jacobi identity in place of Ripa’s long outline proof. The same bracket has appeared before in compressible and relativistic magnetohydrodynamics. The Hamiltonian is the integral of the three dimensional energy density for both the inhomogeneous layer and magnetohydrodynamic systems, which provides a compact derivation of Ripa’s models.

Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics

Shallow water magnetohydrodynamics is a recently proposed model for a thin layer of incompressible, electrically conducting fluid. The velocity and magnetic field are taken to be nearly two dimensional, with approximate magnetohydrostatic balance in the perpendicular direction. In this paper a Hamiltonian description, with the ubiquitous noncanonical Lie–Poisson bracket for barotropic magnetohydrodynamics, is derived by integrating the three-dimensional energy density in the perpendicular direction. Specialization to two dimensions yields an elegant form of the bracket, from which further conserved quantities (Casimirs) of shallow water magnetohydrodynamics are derived. These Casimirs closely resemble the Casimirs of incompressible reduced magnetohydrodynamics, so the stability properties of the two systems may be expected to be similar. The shallow water magnetohydrodynamics system is also cast into symmetric hyperbolic form. The symmetric and Hamiltonian properties become incompatible when the relevant divergence-free constraint ∇⋅(hB)=0 is relaxed.

The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices

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 Föppl equilibrium solutions using the energy-Casimir method.

The twisted top

We describe a new type of top, the twisted top, obtained by appending a cocycle to the Lie–Poisson bracket for the charged heavy top, thus breaking its semidirect product structure. The twisted top has an integrable case that corresponds to the Lagrange (symmetric) top. We give a canonical description of the twisted top in terms of Euler angles. We also show by a numerical calculation of the largest Lyapunov exponent that the Kovalevskaya case of the twisted top is chaotic.