Euler-Poincar\'e approaches to nematodynamics

Nematodynamics is the orientation dynamics of flowless liquid-crystals. We show how Euler-Poincar\'e reduction produces a unifying framework for various theories, including Ericksen-Leslie, Luhiller-Rey, and Eringen's micropolar theory. In particular, we show that these theories are all compatible with each other and some of them allow for more general configurations involving a non vanishing discination density. All results are also extended to flowing liquid crystals.Comment: 26 pages, no figure

A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

A variational principle is derived for two-dimensional incompressible rotational fluid flow with a free surface in a moving vessel when both the vessel and fluid motion are to be determined. The fluid is represented by a stream function and the vessel motion is represented by a path in the planar Euclidean group. Novelties in the formulation include how the pressure boundary condition is treated, the introduction of a stream function into the Euler-Poincar\'e variations, the derivation of free surface variations, and how the equations for the vessel path in the Euclidean group, coupled to the fluid motion, are generated automatically.Comment: 19 pages, 3 figure

Variational integrators for anelastic and pseudo-incompressible flows

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincaré variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighted velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases

Variational principles for spin systems and the Kirchhoff rod

We obtain the affine Euler-Poincar\'e equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.Comment: Submitted to Journal of Geometric Mechanics, 29 pages, no figure

Geometric analysis of noisy perturbations to nonholonomic constraints

We propose two types of stochastic extensions of nonholonomic constraints for mechanical systems. Our approach relies on a stochastic extension of the Lagrange-d'Alembert framework. We consider in details the case of invariant nonholonomic systems on the group of rotations and on the special Euclidean group. Based on this, we then develop two types of stochastic deformations of the Suslov problem and study the possibility of extending to the stochastic case the preservation of some of its integrals of motion such as the Kharlamova or Clebsch-Tisserand integrals

Validity domain and limitation of non-retarded Green's tensor for electromagnetic scattering at surfaces

This work gives a detailed derivation of the non-retarded dyadic Green's tensor associated with surfaces in the quasistatic approximation. The derivation is made from a rigorous model where the dyadic is expressed as Sommerfeld integrals. We then assess the domain where this approximation can be used for scattering calculations on surfaces by comparing rigorous and non-retarded solutions. Implications of this work for scattering calculations in near-field optics are finally discussed. (C) 2000 Elsevier Science B.V. All rights reserved

Electromagnetic scattering of high-permittivity particles on a substrate

We contribute to the study of the optical properties of high-permittivity nanostructures deposited on surfaces. We present what we believe is anew computational technique derived from the coupled-dipole approximation (CDA), which can accommodate high-permittivity scatterers. The discretized CDA equations are reformulated by use of the sampling theory to overcome different sources of inaccuracy that arise for high-permittivity scatterers. We first give the nonretarded filtered surface Green's tensor used in the new scheme. We then assess the accuracy of the technique by comparing it with the standard CDA approach and show that it can accurately handle scatterers with a large permittivity, (C) 2001 Optical Society of America

A library for computing the filtered and non-filtered 3D Green's tensor associated with infinite homogeneous space and surfaces

We describe a library to compute various types of Green's tensor for three-dimensional electromagnetic scattering calculations. This library includes the retarded and non-retarded (quasi-static) Green's tensors for infinite homogeneous space and the non-retarded Green's tensor associated with a surface. Both standard and filtered Green's tensor can be computed. Filtered Green's tensor can be used to accurately investigate high permittivity scatterers with the coupled-dipole approximation. (C) 2002 Elsevier Science B.V. All rights reserved

Accurate and efficient computation of the Green's tensor for stratified media

We present a technique for the computation of the Green's tensor in three-dimensional stratified media composed of an arbitrary number of layers with different permittivities and permeabilities (including metals with a complex permittivity). The practical implementation of this technique is discussed in detail. In particular, we show how to efficiently handle the singularities occurring in Sommerfeld integrals, by deforming the integration path in the complex plane. Examples assess the accuracy of this approach and illustrate the physical properties of the Green's tensor, which represents the field radiated by three orthogonal dipoles embedded in the multilayered medium