Lie Groups and mechanics: an introduction

The aim of this paper is to present aspects of the use of Lie groups in mechanics. We start with the motion of the rigid body for which the main concepts are extracted. In a second part, we extend the theory for an arbitrary Lie group and in a third section we apply these methods for the diffeomorphism group of the circle with two particular examples: the Burger equation and the Camassa-Holm equation

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

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

The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and dym type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.Comment: 31 pages, no figures, to appear in Commun. Math. Phy

Killed in action (KIA): an analysis of military personnel who died of their injuries before reaching a definitive medical treatment facility in Afghanistan (2004-2014).

INTRODUCTION: The majority of combat deaths occur before arrival at a medical treatment facility but no previous studies have comprehensively examined this phase of care. METHODS: The UK Joint Theatre Trauma Registry was used to identify all UK military personnel who died in Afghanistan (2004-2014). These data were linked to non-medical tactical and operational records to provide an accurate timeline of events. Cause of death was determined from records taken at postmortem review. The primary objective was to report time between injury and death in those killed in action (KIA); secondary objectives included: reporting mortality at key North Atlantic Treaty Organisation timelines (0, 10, 60, 120 min), comparison of temporal lethality for different anatomical injuries and analysing trends in the case fatality rate (CFR). RESULTS: 2413 UK personnel were injured in Afghanistan from 2004 to 2014; 448 died, with a CFR of 18.6%. 390 (87.1%) of these died prehospital (n=348 KIA, n=42 killed non-enemy action). Complete data were available for n=303 (87.1%) KIA: median Injury Severity Score 75.0 (IQR 55.5-75.0). The predominant mechanisms were improvised explosive device (n=166, 54.8%) and gunshot wound (n=96, 31.7%).In the KIA cohort, the median time to death was 0.0 (IQR 0.0-21.8) min; 173 (57.1%) died immediately (0 min). At 10, 60 and 120 min post injury, 205 (67.7%), 277 (91.4%) and 300 (99.0%) casualties were dead, respectively. Whole body primary injury had the fastest mortality. Overall prehospital CFR improved throughout the period while in-hospital CFR remained constant. CONCLUSION: Over two-thirds of KIA deaths occurred within 10 min of injury. Improvement in the CFR in Afghanistan was predominantly in the prehospital phase

Geodesic Warps by Conformal Mappings

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g.\ in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations

Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Sarcoidosis: the disease and its ocular manifestations

Sarcoidosis is a systemic disease that has ocular manifestations. For many patients there is a long course of treatment both topical and systemic. This article describes sarcoidosis both systemic and ocular and some of the treatment modalities used