Routh reduction and the class of magnetic Lagrangian systems

In this paper, some new aspects related to Routh reduction of Lagrangian systems with symmetry are discussed. The main result of this paper is the introduction of a new concept of transformation that is applicable to systems obtained after Routh reduction of Lagrangian systems with symmetry, so-called magnetic Lagrangian systems. We use these transformations in order to show that, under suitable conditions, the reduction with respect to a (full) semi-direct product group is equivalent to the reduction with respect to an Abelian normal subgroup. The results in this paper are closely related to the more general theory of Routh reduction by stages.Comment: 23 page

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

Nonaffine Correlations in Random Elastic Media

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements \uv (\xv) from a uniform reference state grow linearly with position \xv, and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements \uv' (\xv) as the difference between \uv (\xv) and affine displacements, and we investigate the nonaffinity correlation function $\mathcal{G} =$ and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants, by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that $\mathcal{G}$ scales as A |\xv|^{-(d-2)} where the amplitude $A$ is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short and long-range correlations in random elastic moduli.Comment: 22 Pages, 18 figures, RevTeX

A millimeter-wave antireflection coating for cryogenic silicon lenses

We have developed and tested an antireflection (AR) coating method for silicon lenses at cryogenic temperatures and millimeter wavelengths. Our particular application is a measurement of the cosmic microwave background. The coating consists of machined pieces of Cirlex glued to the silicon. The measured reflection from an AR coated flat piece is less than 1.5% at the design wavelength. The coating has been applied to flats and lenses and has survived multiple thermal cycles from 300 to 4 K. We present the manufacturing method, the material properties, the tests performed, and estimates of the loss that can be achieved in practical lenses

Routhian reduction for quasi-invariant Lagrangians

In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We show how functional Routhian reduction can be seen as a particular instance of reduction of a quasi-invariant Lagrangian, and we exhibit a Routhian reduction procedure for the special case of Lagrangians with quasi-cyclic coordinates. As an application we consider the dynamics of a charged particle in a magnetic field.Comment: 24 pages, 3 figure

Discrete Lie Advection of Differential Forms

In this paper, we present a numerical technique for performing Lie advection of arbitrary differential forms. Leveraging advances in high-resolution finite volume methods for scalar hyperbolic conservation laws, we first discretize the interior product (also called contraction) through integrals over Eulerian approximations of extrusions. This, along with Cartan's homotopy formula and a discrete exterior derivative, can then be used to derive a discrete Lie derivative. The usefulness of this operator is demonstrated through the numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC

Spacelike surfaces with free boundary in the Lorentz-Minkowski space

We investigate a variational problem in the Lorentz-Minkowski space \l^3 whose critical points are spacelike surfaces with constant mean curvature and making constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap. We also study the problem of spacelike hypersurfaces with free boundary in the higher dimensional Lorentz-Minkowski space \l^{n+1}.Comment: 16 pages. Accepted in Classical and Quantum Gravit

Winds of Planet Hosting Stars

The field of exoplanetary science is one of the most rapidly growing areas of astrophysical research. As more planets are discovered around other stars, new techniques have been developed that have allowed astronomers to begin to characterise them. Two of the most important factors in understanding the evolution of these planets, and potentially determining whether they are habitable, are the behaviour of the winds of the host star and the way in which they interact with the planet. The purpose of this project is to reconstruct the magnetic fields of planet hosting stars from spectropolarimetric observations, and to use these magnetic field maps to inform simulations of the stellar winds in those systems using the Block Adaptive Tree Solar-wind Roe Upwind Scheme (BATS-R-US) code. The BATS-R-US code was originally written to investigate the behaviour of the Solar wind, and so has been altered to be used in the context of other stellar systems. These simulations will give information about the velocity, pressure and density of the wind outward from the host star. They will also allow us to determine what influence the winds will have on the space weather environment of the planet. This paper presents the preliminary results of these simulations for the star $\tau$ Bo\"otis, using a newly reconstructed magnetic field map based on previously published observations. These simulations show interesting structures in the wind velocity around the star, consistent with the complex topology of its magnetic field.Comment: 8 pages, 2 figures, accepted for publication in the peer-reviewed proceedings of the 14th Australian Space Research Conference, held at the University of South Australia, 29th September - 1st October 201

On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets

The role of projectors associated with Poisson brackets of constrained Hamiltonian systems is analyzed. Projectors act in two instances in a bracket: in the explicit dependence on the variables and in the computation of the functional derivatives. The role of these projectors is investigated by using Dirac's theory of constrained Hamiltonian systems. Results are illustrated by three examples taken from plasma physics: magnetohydrodynamics, the Vlasov-Maxwell system, and the linear two-species Vlasov system with quasineutrality

Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials

It is demonstrated numerically that smooth three degrees of freedom Hamiltonian systems which are arbitrarily close to three dimensional strictly dispersing billiards (Sinai billiards) have islands of effective stability, and hence are non-ergodic. The mechanism for creating the islands are corners of the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao