Gauge Theory for Finite-Dimensional Dynamical Systems

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory

Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines

We propose a class of electrical circuits for extremely wideband (EWB) signal shaping. A one-dimensional, nonlinear, nonuniform transmission line is proposed for narrow pulse generation. A two-dimensional transmission lattice is proposed for EWB signal combining. Model equations for the circuits are derived. Theoretical and numerical solutions of the model equations are presented, showing that the circuits can be used for the desired application. The procedure by which the circuits are designed exemplifies a modern, mathematical design methodology for EWB circuits

Lagrangian Reduction, the Euler--Poincar\'{e} 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.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure

Fuzzy Fluid Mechanics in Three Dimensions

We introduce a rotation invariant short distance cut-off in the theory of an ideal fluid in three space dimensions, by requiring momenta to take values in a sphere. This leads to an algebra of functions in position space is non-commutative. Nevertheless it is possible to find appropriate analogues of the Euler equations of an ideal fluid. The system still has a hamiltonian structure. It is hoped that this will be useful in the study of possible singularities in the evolution of Euler (or Navier-Stokes) equations in three dimensions.Comment: Additional reference

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

Design and synthesis of a fragment set based on twisted bicyclic lactams

Current fragment sets tend to be dominated by flatter molecules, and their shape diversity does not reflect that of the fragments that are theoretically possible. The design and synthesis of a set of bridged fragments containing a bridgehead nitrogen is described. Many of these fragments contain twisted lactams whose modulated electronic properties may present unusual opportunities for interaction with target proteins. The demonstrated novelty, three-dimensionality and molecular properties of the set of 22 fragments may provide valuable, and highly distinctive, starting points for fragment-based drug discovery

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

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

Factors associated with limited exercise capacity and feasibility of high intensity interval training in people with mild to moderate Parkinson's disease

Background/Aims: Fitness and function can improve with exercise in people with Parkinson's disease. Animal models suggest that exercise may also have a neuroprotective effect, with higher intensity exercise being more beneficial than lower intensity exercise. However, in people with Parkinson's disease the factors limiting exercise capacity are not fully understood and it is unclear whether training at very high intensities would be safe, feasible and acceptable. Methods: Eighteen people with Parkinson's disease were recruited to explore respiratory and neuromuscular factors that may limit exercise capacity. In a purposive subgroup of 6 participants able to achieve &gt;75% of their predicted maximum heart rate the feasibility of undertaking six high intensity interval training sessions over 3 weeks was tested. Their experience was further explored in a focus group. Results: Lower exercise capacity was associated with lower limb flexor muscle strength (r2=0.51) but not with disease severity or respiratory function. There were no adverse events or drop-outs in those taking part in the exercise regimen. Improvements were seen in fitness, health related quality of life, activity levels, walking speed, muscle strength and cycle endurance. Participants reported that they enjoyed high intensity, supervised exercise. High intensity interval training may be feasible and safe. Conclusions: We concluded that high intensity interval training has the potential to be a safe and acceptable mode of exercise in this patient group. </jats:sec