752 research outputs found
Invariants and Labels in Lie-Poisson Systems
Reduction is a process that uses symmetry to lower the order of a Hamiltonian
system. The new variables in the reduced picture are often not canonical: there
are no clear variables representing positions and momenta, and the Poisson
bracket obtained is not of the canonical type. Specifically, we give two
examples that give rise to brackets of the noncanonical Lie-Poisson form: the
rigid body and the two-dimensional ideal fluid. From these simple cases, we
then use the semidirect product extension of algebras to describe more complex
physical systems. The Casimir invariants in these systems are examined, and
some are shown to be linked to the recovery of information about the
configuration of the system. We discuss a case in which the extension is not a
semidirect product, namely compressible reduced MHD, and find for this case
that the Casimir invariants lend partial information about the configuration of
the system.Comment: 11 pages, RevTeX. To appear in Proceedings of the 13th Florida
Workshop in Astronomy and Physic
Singular Casimir Elements of the Euler Equation and Equilibrium Points
The problem of the nonequivalence of the sets of equilibrium points and
energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian
formulation of equations describing ideal fluid and plasma dynamics, is
addressed in the context of the Euler equation for an incompressible inviscid
fluid. The problem is traced to a Casimir deficit, where Casimir elements
constitute the center of the Lie-Poisson algebra underlying the Hamiltonian
formulation, and this leads to a study of the symplectic operator defining the
Poisson bracket. The kernel of the symplectic operator, for this typical
example of an infinite-dimensional Hamiltonian system for media in terms of
Eulerian variables, is analyzed. For two-dimensional flows, a rigorously
solvable system is formulated. The nonlinearity of the Euler equation makes the
symplectic operator inhomogeneous on phase space (the function space of the
state variable), and it is seen that this creates a singularity where the
nullity of the symplectic operator (the "dimension" of the center) changes.
Singular Casimir elements stemming from this singularity are unearthed using a
generalization of the functional derivative that occurs in the Poisson bracket
Classification and Casimir Invariants of Lie-Poisson Brackets
We classify Lie-Poisson brackets that are formed from Lie algebra extensions.
The problem is relevant because many physical systems owe their Hamiltonian
structure to such brackets. A classification involves reducing all brackets to
a set of normal forms, and is achieved partially through the use of Lie algebra
cohomology. For extensions of order less than five, the number of normal forms
is small and they involve no free parameters. We derive a general method of
finding Casimir invariants of Lie-Poisson bracket extensions. The Casimir
invariants of all low-order brackets are explicitly computed. We treat in
detail a four field model of compressible reduced magnetohydrodynamics.Comment: 59 pages, Elsevier macros. To be published in Physica
Hamiltonian fluid closures of the Vlasov-Ampère equations: from water-bags to N moment models
International audienceMoment closures of the Vlasov-Ampère system, whereby higher moments are represented as functions of lower moments with the constraint that the resulting fluid system remains Hamiltonian, are investigated by using water-bag theory. The link between the water-bag formalism and fluid models that involve density, fluid velocity, pressure and higher moments is established by introducing suitable thermodynamic variables. The cases of one, two and three water-bags are treated and their Hamiltonian structures are provided. In each case, we give the associated fluid closures and we discuss their Casimir invariants. We show how the method can be extended to an arbitrary number of fields, i.e., an arbitrary number of water-bags and associated moments. The thermodynamic interpretation of the resulting models is discussed. Finally, a general procedure to derive Hamiltonian N-field fluid models is proposed
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Relabeling symmetries in hydrodynamics and magnetohydrodynamics
Lagrangian symmetries and concomitant generalized Bianchi identities associated with the relabeling of fluid elements are found for hydrodynamics and magnetohydrodynamics (MHD). In hydrodynamics relabeling results in Ertel`s theorem of conservation of potential vorticity, while in MHD it yields the conservation of cross helicity. The symmetries of the reduction from Lagrangian (material) to Eulerian variables are used to construct the Casimir invariants of the Hamiltonian formalism
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Nonlinear instability and chaos in plasma wave-wave interactions. II. Numerical methods and results
In Part I of this work and Physics of Plasmas, June 1995, the behavior of linearly stable, integrable systems of waves in a simple plasma model was described using a Hamiltonian formulation. It was shown that explosive instability arises from nonlinear coupling between modes of positive and negative energy, with well-defined threshold amplitudes depending on the physical parameters. In this concluding paper, the nonintegrable case is treated numerically. Several sets of waves are considered, comprising systems of two and three degrees of freedom. The time evolution is modelled with an explicit symplectic integration algorithm derived using Lie algebraic methods. When initial wave amplitudes are large enough to support two-wave decay interactions, strongly chaotic motion destroys the separatrix bounding the stable region for explosive triplets. Phase space orbits then experience diffusive growth to amplitudes that are sufficient for explosive instability, thus effectively reducing the threshold amplitude. For initial amplitudes too small to drive decay instability, small perturbations might still grow to arbitrary size via Arnold diffusion. Numerical experiments do not show diffusion in this case, although the actual diffusion rate is probably underestimated due to the simplicity of the model
Chern-Simons Reduction and non-Abelian Fluid Mechanics
We propose a non-Abelian generalization of the Clebsch parameterization for a
vector in three dimensions. The construction is based on a group-theoretical
reduction of the Chern-Simons form on a symmetric space. The formalism is then
used to give a canonical (symplectic) discussion of non-Abelian fluid
mechanics, analogous to the way the Abelian Clebsch parameterization allows a
canonical description of conventional fluid mechanics.Comment: 12 pages, REVTeX; revised for publication in Phys Rev D; email to
[email protected]
What a classical r-matrix really is
The notion of classical -matrix is re-examined, and a definition suitable
to differential (-difference) Lie algebras, -- where the standard definitions
are shown to be deficient, -- is proposed, the notion of an -operator. This notion has all the natural properties one would expect form
it, but lacks those which are artifacts of finite-dimensional isomorpisms such
as not true in differential generality relation \mbox{End}\, (V) \simeq V^*
\otimes V for a vector space . Examples considered include a quadratic
Poisson bracket on the dual space to a Lie algebra; generalized
symplectic-quadratic models of such brackets (aka Clebsch representations); and
Drinfel'd's 2-cocycle interpretation of nondegenate classical -matrices
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