Geometric analysis of optical frequency conversion and its control in quadratic nonlinear media

We analyze frequency conversion and its control among three light waves using a geometric approach that enables the dynamics of the waves to be visualized on a closed surface in three dimensions. It extends the analysis based on the undepleted-pump linearization and provides a simple way to understand the fully nonlinear dynamics. The Poincaré sphere has been used in the same way to visualize polarization dynamics. A geometric understanding of control strategies that enhance energy transfer among interacting waves is introduced, and the quasi-phase-matching strategy that uses microstructured quadratic materials is illustrated in this setting for both type I and II second-harmonic generation and for parametric three-wave interactions

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo

Transport and Older People: Integrating Transport Planning Tools with User Needs

This study was funded through a pump-priming grant from the Strategic Promotion of Ageing Research Capacity (SPARC) programme. The purpose of the project was to bring together transport and public health research in order to demonstrate how the involvement of older people can help improve tools for transport planning. The study was unique in that it brought together public health and transport planning and engineering with older people to consider how services can be more responsive to older people’s transport needs. The project had five research objectives: 1. To investigate how accessibility problems impact on older people’s independence 2. To determine the extent to which currently available data sources and modelling tools reflect older people’s stated accessibility needs 3. To understand how the gap between expected and perceived accessibility problems varies across different categories of older people 4. To pilot techniques that could be applied to provide a more robust measure of accessibility for older people. 5. To build new research capacity across disciplines to develop a national focus on the interactions between ageing and transport planning. The methods were determined on the basis of ‘appropriate tools with maximum output’. Focus group interviews were selected as a useful tool for reaching a large number of older people within a limited time span, for providing an arena for discussion and debate about a topical subject and for generating ideas for improving transport planning. Following the interviews accompanied walks were undertaken with older people in a range of road environments and traffic situations. The purpose of these walks was to observe and explore the way older people interact with their environment. Data from the focus group interviews and the observations were compared with the outputs from an accessibility planning tool used by local authorities to plan accessible and acceptable transport routes (Accession™). The purpose of this exercise was to investigate whether or not such tools are able to take into account the varying needs of older people. The study was undertaken over eight months. Eighty one older people living in the Leeds district took part in the focus groups. They covered a broad range of mobility levels and used a variety of transport types, as such a reasonably rounded perspective on the issues concerned was offered. In addition six walks were undertaken with older people in their community

Dirac reduction revisited

The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu reduction procedure. The Dirac classification of constraints on 'first-class' constraints and 'second-class' constraints is reexamined.Comment: This is a revised version of an article published in J. Nonlinear Math. Phys. vol. 10, No. 4, (2003), 451-46

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

Cometary Astrometry

Modern techniques for making cometary astrometric observations, reducing these observations, using accurate reference star catalogs, and computing precise orbits and ephemerides are discussed in detail and recommendations and suggestions are given in each area

Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group

In this work we study some symplectic submanifolds in the cotangent bundle of a factorizable Lie group defined by second class constraints. By applying the Dirac method, we study many issues of these spaces as fundamental Dirac brackets, symmetries, and collective dynamics. This last item allows to study integrability as inherited from a system on the whole cotangent bundle, leading in a natural way to the AKS theory for integrable systems

The quasi-bi-Hamiltonian formulation of the Lagrange top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the possible reductions of the Poisson tensors, the vector field and its Hamiltonian functions on a four-dimensional space. We show that the vector field of the Lagrange top possesses, on the reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen. (March 2002

Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design

The invariant manifold structures of the collinear libration points for the spatial restricted three-body problem provide the framework for understanding complex dynamical phenomena from a geometric point of view. In particular, the stable and unstable invariant manifold \tubes" associated to libration point orbits are the phase space structures that provide a conduit for orbits between primary bodies for separate three-body systems. These invariant manifold tubes can be used to construct new spacecraft trajectories, such as a \Petit Grand Tour" of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. The current work extends the results to the spatial case

Symmetry Reduction by Lifting for Maps

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure