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

Tracking implicit trajectories

Output tracking of implcitly defined reference trajectories is examined. A continuous-time nonlinear dynamical system is constructed that produces explicit estimates of time-varying implicit trajectories. We prove that incorporation of this "dynamic inverter" into a tracking controller provides exponential output tracking of the implicitly defined trajectory for nonlinear control systems having vector relative degree and well-behaved internal dynanmics

Control for an Autonomous Bicycle

The control of nonholonomic and underactuated systems with symmetry is illustrated by the problem of controlling a bicycle. We derive a controller which, using steering and rear-wheel torque, causes a model of a riderless bicycle to recover its balance from a near fall as well as converge to a time parameterized path in the ground plane. Our construction utilizes new results for both the derivation of equations of motion for nonholonomic systems with symmetry, as well as the control of underactuated robotic systems

Joint-space tracking of workspace trajectories in continuous time

We present a controller for a class of robotics manipulators which provides exponential convergence to a desired end-effector trajectory using gains specified in joint-space. This is accomplished without appeal to the use of discrete inverse-kinematics algorithms, allowing the controller to be posed entirely in continuous time

Dynamic inversion and polar decomposition of matrices

Using the recently introduced concept of a "dynamic inverse" of a map, along with its associated analog computational paradigm. we construct continuous-time nonlinear dynamical systems which produce both regular and generalized inverses of time-varying and fixed matrices, as well as polar decompositions

A dynamic inverse for nonlinear maps

We consider the problem of estimating the time-varying root of a time-dependent nonlinear map. We introduce a "dynamic inverse" of a map, another generally time-dependent map which one composes with the original map to form a nonlinear vector-field. The flow of this vector field decays exponentially to the root. We then show how a dynamic inverse may be determined dynamically while being used simultaneously to find a root. We construct a continuous-time analog computational paradigm around the dynamic inverse

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

Kiss and Fly - a study of the impacts at a UK regional airport

In the light of the forecast growth in air transport the UK Government has placed a requirement on all airports with substantial air transport movements to implement surface access strategies. The emphasis of surface access policy has been to increase the proportion of people arriving at airports by public transport by a variety of means such as managing parking supply and pricing and improving public transport. The extent to which these policies will be effective will depend on a number of factors such as the quality and availability of the alternatives, the availability of competing off-site parking and the extent to which kiss and fly is feasible. This paper reports on two studies of passenger access to Leeds-Bradford International Airport in the summers of 2004 and 2005. The airport has an aspiration to increase public transport use to the airport from its current level of 3% to 10% by 2010. The principal means by which this is currently planned to be achieved is through the expansion of scheduled bus services to Leeds, Bradford and Harrogate. The 2004 study found that 49% of passengers were dropped off at the airport by friends and that the potential for larger quantities of people to reach the airport by conventional bus services was limited. The 2005 study investigated the extent to which these kiss and fly journeys generate extra travel on the road network. The results show that for an airport with around 2.5 million passengers the Kiss & Fly journeys are creating an extra 19.4 million kilometres, an increase of 36% over the distance that would have been travelled if people had driven and parked. The paper concludes that a charge levied on all vehicles accessing the airport, similar to a congestion charge, is likely to have the greatest impact on travel behaviour and will have a far greater impact on the environment than the current emphasis on public transport improvements and parking restrictions

On uniformly rotating fluid drops trapped between two parallel plates

This contribution is about the dynamics of a liquid bridge between two fixed parallel plates. We consider a mathematical model and present some results from the doctoral thesis [10] of the first author. He showed that there is a Poisson bracket and a corresponding Hamiltonian, so that the model equations are in Hamiltonian form. The result generalizes previous results of Lewis et al. on the dynamics of free boundary problems for "free" liquid drops to the case of a drop between two parallel plates, including, especially the effect of capillarity and the angle of contact between the plates and the free fluid surface. Also, we prove the existence of special solutions which represent uniformly rotating fluid ridges, and we present specific stability conditions for these solutions. These results extend work of Concus and Finn [2] and Vogel [18],[19] on static capillarity problems (see also Finn [5]). Using the Hamiltonian structure of the model equations and symmetries of the solutions, the stability conditions can be derived in a systematic way. The ideas that are described will be useful for other situations involving capillarity and free boundary problems as well