Topics in the Dynamics or General Relativity

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The Einstein Equations of Evolution - A Geometric Approach

In this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite-dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?" By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body-space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time-dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the "intrinsic shift vector field." The essence of our method is to extend the usual configuration space [fraktur M]=Riem(M) of Riemannian metrics to [script T]×[script D]×[fraktur M], where [script T]=C[infinity](M,R) is the group of relativistic time translations and [script D]=Diff(M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields (xit,etat)[is-an-element-of][script T]×[script D]. On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for [fraktur M] as a degenerate metric on [script D]×[fraktur M]. On [script D]×[fraktur M], however, the metric is quadratic in the velocity variables. The groups [script T] and [script D] also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual "constraint equations." A precise theorem is given for a remark of Misner that in an empty space-time we must have [script H]=0. We study the relationship between the evolution equations for the time-dependent metric gt and the Ricci flat condition of the reconstructed Lorentz metric gL. Finally, we make some remarks about a possible "superphase space" for general relativity and how our treatment on [script T]×[script D]×[fraktur M] is related to ordinary superspace and superphase space

Control and stabilization of systems with homoclinic orbits

In this paper we consider the control of two physical systems, the near wall region of a turbulent boundary layer and the rigid body, using techniques from the theory of nonlinear dynamical systems. Both these systems have saddle points linked by heteroclinic orbits. In the fluid system we show how the structure of the phase space can be used to keep the system near an (unstable) saddle. For the rigid body system we discuss passage along the orbit as a possible control manouver, and show how the Energy-Casimir method can be used to analyze stabilization of the system about the saddles

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

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

Physical Dissipation and the Method of Controlled Lagrangians

We describe the effect of physical dissipation on stability of equilibria which have been stabilized, in the absence of damping, using the method of controlled Lagrangians. This method applies to a class of underactuated mechanical systems including “balance” systems such as the pendulum on a cart. Since the method involves modifying a system’s kinetic energy metric through feedback, the effect of dissipation is obscured. In particular, it is not generally true that damping makes a feedback-stabilized equilibrium asymptotically stable. Damping in the unactuated directions does tend to enhance stability, however damping in the controlled directions must be “reversed” through feedback. In this paper, we suggest a choice of feedback dissipation to locally exponentially stabilize a class of controlled Lagrangian systems