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 $Z_2$-Regge model could be such a desired simplification. Here the quadratic edge lengths $q$ of the simplicial complexes are restricted to only two possible values $q=1+\epsilon\sigma$, with $\sigma=\pm 1$, 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