5,375 research outputs found
Variational Principles for Lagrangian Averaged Fluid Dynamics
The Lagrangian average (LA) of the ideal fluid equations preserves their
transport structure. This transport structure is responsible for the Kelvin
circulation theorem of the LA flow and, hence, for its convection of potential
vorticity and its conservation of helicity.
Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure
A block diagonalization theorem in the energy-momentum method
We prove a geometric generalization of a block diagonalization theorem first found by the authors for
rotating elastic rods. The result here is given in the general context of simple mechanical systems with a
symmetry group acting by isometries on a configuration manifold. The result provides a choice of
variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of
an augmented energy these variables effectively separate the rotational and internal vibrational modes. The
second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the
symplectic form has an off diagonal term which represents the dynamic interaction between these modes.
Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development
of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques
should apply to other systems as well, such as rotating fluid masses
Normalizing connections and the energy-momentum method
The block diagonalization method for determining the stability of relative equilibria is discussed from
the point of view of connections. We construct connections whose horizontal and vertical decompositions simultaneosly put the second variation of the augmented Hamiltonian and the symplectic structure into normal form. The cotangent bundle reduction theorem provides the setting in which the results are obtained
Hamiltonian approach to hybrid plasma models
The Hamiltonian structures of several hybrid kinetic-fluid models are
identified explicitly, upon considering collisionless Vlasov dynamics for the
hot particles interacting with a bulk fluid. After presenting different
pressure-coupling schemes for an ordinary fluid interacting with a hot gas, the
paper extends the treatment to account for a fluid plasma interacting with an
energetic ion species. Both current-coupling and pressure-coupling MHD schemes
are treated extensively. In particular, pressure-coupling schemes are shown to
require a transport-like term in the Vlasov kinetic equation, in order for the
Hamiltonian structure to be preserved. The last part of the paper is devoted to
studying the more general case of an energetic ion species interacting with a
neutralizing electron background (hybrid Hall-MHD). Circulation laws and
Casimir functionals are presented explicitly in each case.Comment: 27 pages, no figures. To appear in J. Phys.
Discrete Routh Reduction
This paper develops the theory of abelian Routh reduction for discrete
mechanical systems and applies it to the variational integration of mechanical
systems with abelian symmetry. The reduction of variational Runge-Kutta
discretizations is considered, as well as the extent to which symmetry
reduction and discretization commute. These reduced methods allow the direct
simulation of dynamical features such as relative equilibria and relative
periodic orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth orbiting
satellite with a non-spherical correction, as well as the double
spherical pendulum. The problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to numerical
discretization can be hard to distinguish, but this issue does not appear in
the reduced algorithm, where one can directly observe interesting dynamical
structures in the reduced phase space (the cotangent bundle of shape space), in
which the geometric phases have been removed. The main feature of the double
spherical pendulum example is that it has a nontrivial magnetic term in its
reduced symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic structure. In
contrast, a traditional symplectic method for canonical systems could require
repeated coordinate changes if one is evoking Darboux' theorem to transform the
symplectic structure into canonical form, thereby incurring additional
computational cost. Our method allows one to design reduced symplectic
integrators in a natural way, despite the noncanonical nature of the symplectic
structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added,
fixed typo
Explicit Lie-Poisson integration and the Euler equations
We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson
integrators, preserving all Casimirs, can be constructed. The integrators are
extremely simple. Examples are the rigid body, a moment truncation, and a new,
fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho
The Hamiltonian structure and Euler-Poincar\'{e} formulation of the Vlasov-Maxwell and gyrokinetic systems
We present a new variational principle for the gyrokinetic system, similar to
the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in
the Eulerian frame and based on constrained variations of the phase space fluid
velocity and particle distribution function. Using a Legendre transform, we
explicitly derive the field theoretic Hamiltonian structure of the system. This
is carried out with a modified Dirac theory of constraints, which is used to
construct meaningful brackets from those obtained directly from
Euler-Poincar\'{e} theory. Possible applications of these formulations include
continuum geometric integration techniques, large-eddy simulation models and
Casimir type stability methods.
[1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)Comment: 36 pages, 1 figur
Is the electrostatic force between a point charge and a neutral metallic object always attractive?
We give an example of a geometry in which the electrostatic force between a
point charge and a neutral metallic object is repulsive. The example consists
of a point charge centered above a thin metallic hemisphere, positioned concave
up. We show that this geometry has a repulsive regime using both a simple
analytical argument and an exact calculation for an analogous two-dimensional
geometry. Analogues of this geometry-induced repulsion can appear in many other
contexts, including Casimir systems.Comment: 7 pages, 7 figure
Bulgac-Kusnezov-Nos\'e-Hoover thermostats
In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in
phase space by means of non-Hamiltonian brackets. Two generalized versions of
the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are
globally controlled by means of a single additional Nos\'e variable, and
another where each demon is coupled to an independent Nos\'e-Hoover thermostat.
Numerically stable and efficient measure-preserving time-reversible algorithms
are derived in a systematic way for each case. The chaotic properties of the
different phase space flows are numerically illustrated through the
paradigmatic example of the one-dimensional harmonic oscillator. It is found
that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic,
both of the Nos\'e-Hoover controlled dynamics sample the canonical distribution
correctly
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