Lagrangian reduction, the Euler--Poincaré Equations, and semidirect products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given

Assessing architectural evolution: A case study

This is the post-print version of the Article. The official published can be accessed from the link below - Copyright @ 2011 SpringerThis paper proposes to use a historical perspective on generic laws, principles, and guidelines, like Lehman’s software evolution laws and Martin’s design principles, in order to achieve a multi-faceted process and structural assessment of a system’s architectural evolution. We present a simple structural model with associated historical metrics and visualizations that could form part of an architect’s dashboard. We perform such an assessment for the Eclipse SDK, as a case study of a large, complex, and long-lived system for which sustained effective architectural evolution is paramount. The twofold aim of checking generic principles on a well-know system is, on the one hand, to see whether there are certain lessons that could be learned for best practice of architectural evolution, and on the other hand to get more insights about the applicability of such principles. We find that while the Eclipse SDK does follow several of the laws and principles, there are some deviations, and we discuss areas of architectural improvement and limitations of the assessment approach

The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of ${\mathbb R}^n$, this system of PDEs with Dirichlet boundary conditions are well-posed for initial data in the Hilbert space $H^s$, $s>n/2+1$. We then use a nonlinear Trotter product formula to prove that solutions of the averaged Euler equations are a regular limit of solutions to the averaged Navier-Stokes equations in the limit of zero viscosity. This system of PDEs is also the model for second-grade non-Newtonian fluids

Equivalent theories of liquid crystal dynamics

There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the micropolar theory comprises the director theory. In this paper we show that this is the case by using symmetry reduction techniques and introducing a new system that is equivalent to the Ericksen-Leslie equations and includes disclination dynamics. The resulting equations of motion are verified to be completely equivalent, although one of the two different reductions offers the possibility of accounting for orientational defects. After applying these two approaches to the ordered micropolar theory of Lhuiller and Rey, all the results are eventually extended to flowing complex fluids, such as nematic liquid crystals

Equivalent theories of liquid crystal dynamics

There are two competing descriptions of nematic liquid crystal dynamics: the Ericksen-Leslie director theory and the Eringen micropolar approach. Up to this day, these two descriptions have remained distinct in spite of several attempts to show that the micropolar theory comprises the director theory. In this paper we show that this is the case by using symmetry reduction techniques and introducing a new system that is equivalent to the Ericksen-Leslie equations and includes disclination dynamics. The resulting equations of motion are verified to be completely equivalent, although one of the two different reductions offers the possibility of accounting for orientational defects. After applying these two approaches to the ordered micropolar theory of Lhuiller and Rey, all the results are eventually extended to flowing complex fluids, such as nematic liquid crystals