11,707 research outputs found
An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid Flow
In this paper we consider the Hamiltonian formulation of the equations of
incompressible ideal fluid flow from the point of view of optimal control
theory. The equations are compared to the finite symmetric rigid body equations
analyzed earlier by the authors. We discuss various aspects of the Hamiltonian
structure of the Euler equations and show in particular that the optimal
control approach leads to a standard formulation of the Euler equations -- the
so-called impulse equations in their Lagrangian form. We discuss various other
aspects of the Euler equations from a pedagogical point of view. We show that
the Hamiltonian in the maximum principle is given by the pairing of the
Eulerian impulse density with the velocity. We provide a comparative discussion
of the flow equations in their Eulerian and Lagrangian form and describe how
these forms occur naturally in the context of optimal control. We demonstrate
that the extremal equations corresponding to the optimal control problem for
the flow have a natural canonical symplectic structure.Comment: 6 pages, no figures. To appear in Proceedings of the 39th IEEEE
Conference on Decision and Contro
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
Electrostatic Disorder-Induced Interactions in Inhomogeneous Dielectrics
We investigate the effect of quenched surface charge disorder on
electrostatic interactions between two charged surfaces in the presence of
dielectric inhomogeneities and added salt. We show that in the linear
weak-coupling regime (i.e., by including mean-field and Gaussian-fluctuations
contributions), the image-charge effects lead to a non-zero disorder-induced
interaction free energy between two surfaces of equal mean charge that can be
repulsive or attractive depending on the dielectric mismatch across the
bounding surfaces and the exact location of the disordered charge distribution.Comment: 7 pages, 2 figure
Lattice Models of Quantum Gravity
Standard Regge Calculus provides an interesting method to explore quantum
gravity in a non-perturbative fashion but turns out to be a CPU-time demanding
enterprise. One therefore seeks for suitable approximations which retain most
of its universal features. The -Regge model could be such a desired
simplification. Here the quadratic edge lengths of the simplicial complexes
are restricted to only two possible values , with
, in close analogy to the ancestor of all lattice theories, the
Ising model. To test whether this simpler model still contains the essential
qualities of the standard Regge Calculus, we study both models in two
dimensions and determine several observables on the same lattice size. In order
to compare expectation values, e.g. of the average curvature or the Liouville
field susceptibility, we employ in both models the same functional integration
measure. The phase structure is under current investigation using mean field
theory and numerical simulation.Comment: 4 pages, 1 figure
Averaged Template Matching Equations
By exploiting an analogy with averaging procedures in fluid
dynamics, we present a set of averaged template matching equations.
These equations are analogs of the exact template matching equations
that retain all the geometric properties associated with the diffeomorphismgrou
p, and which are expected to average out small scale features
and so should, as in hydrodynamics, be more computationally efficient
for resolving the larger scale features. Froma geometric point of view,
the new equations may be viewed as coming from a change in norm that
is used to measure the distance between images. The results in this paper
represent first steps in a longer termpro gram: what is here is only
for binary images and an algorithm for numerical computation is not
yet operational. Some suggestions for further steps to develop the results
given in this paper are suggested
Singular solutions of a modified two-component Camassa-Holm equation
The Camassa-Holm equation (CH) is a well known integrable equation describing
the velocity dynamics of shallow water waves. This equation exhibits
spontaneous emergence of singular solutions (peakons) from smooth initial
conditions. The CH equation has been recently extended to a two-component
integrable system (CH2), which includes both velocity and density variables in
the dynamics. Although possessing peakon solutions in the velocity, the CH2
equation does not admit singular solutions in the density profile. We modify
the CH2 system to allow dependence on average density as well as pointwise
density. The modified CH2 system (MCH2) does admit peakon solutions in velocity
and average density. We analytically identify the steepening mechanism that
allows the singular solutions to emerge from smooth spatially-confined initial
data. Numerical results for MCH2 are given and compared with the pure CH2 case.
These numerics show that the modification in MCH2 to introduce average density
has little short-time effect on the emergent dynamical properties. However, an
analytical and numerical study of pairwise peakon interactions for MCH2 shows a
new asymptotic feature. Namely, besides the expected soliton scattering
behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the
phase shift of the peakon collision to diverge in certain parameter regimes.Comment: 25 pages, 11 figure
Equilibrium solutions of the shallow water equations
A statistical method for calculating equilibrium solutions of the shallow
water equations, a model of essentially 2-d fluid flow with a free surface, is
described. The model contains a competing acoustic turbulent {\it direct}
energy cascade, and a 2-d turbulent {\it inverse} energy cascade. It is shown,
nonetheless that, just as in the corresponding theory of the inviscid Euler
equation, the infinite number of conserved quantities constrain the flow
sufficiently to produce nontrivial large-scale vortex structures which are
solutions to a set of explicitly derived coupled nonlinear partial differential
equations.Comment: 4 pages, no figures. Submitted to Physical Review Letter
The Formation and Coarsening of the Concertina Pattern
The concertina is a magnetization pattern in elongated thin-film elements of
a soft material. It is a ubiquitous domain pattern that occurs in the process
of magnetization reversal in direction of the long axis of the small element.
Van den Berg argued that this pattern grows out of the flux closure domains as
the external field is reduced. Based on experimental observations and theory,
we argue that in sufficiently elongated thin-film elements, the concertina
pattern rather bifurcates from an oscillatory buckling mode. Using a reduced
model derived by asymptotic analysis and investigated by numerical simulation,
we quantitatively predict the average period of the concertina pattern and
qualitatively predict its hysteresis. In particular, we argue that the
experimentally observed coarsening of the concertina pattern is due to
secondary bifurcations related to an Eckhaus instability. We also link the
concertina pattern to the magnetization ripple and discuss the effect of a weak
(crystalline or induced) anisotropy
Effects of electromagnetic waves on the electrical properties of contacts between grains
A DC electrical current is injected through a chain of metallic beads. The
electrical resistances of each bead-bead contacts are measured. At low current,
the distribution of these resistances is large and log-normal. At high enough
current, the resistance distribution becomes sharp and Gaussian due to the
creation of microweldings between some beads. The action of nearby
electromagnetic waves (sparks) on the electrical conductivity of the chain is
also studied. The spark effect is to lower the resistance values of the more
resistive contacts, the best conductive ones remaining unaffected by the spark
production. The spark is able to induce through the chain a current enough to
create microweldings between some beads. This explains why the electrical
resistance of a granular medium is so sensitive to the electromagnetic waves
produced in its vicinity.Comment: 4 pages, 5 figure
An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
We study a class of 1+1 quadratically nonlinear water wave equations that
combines the linear dispersion of the Korteweg-deVries (KdV) equation with the
nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still
preserves integrability via the inverse scattering transform (IST) method.
This IST-integrable class of equations contains both the KdV equation and the
CH equation as limiting cases. It arises as the compatibility condition for a
second order isospectral eigenvalue problem and a first order equation for the
evolution of its eigenfunctions. This integrable equation is shown to be a
shallow water wave equation derived by asymptotic expansion at one order higher
approximation than KdV. We compare its traveling wave solutions to KdV
solitons.Comment: 4 pages, no figure
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