Honours Theses' Abstracts

Kappa deformed Special Relativity

John Bowes, 1994.

The theory of deformed symmetries has recently been applied to space-time symmetries in the form of the $\Kappa$-deformed Poincarè algebra, involoving a new fundamental energy scale $\Kappa$.

From the consideration of how this $\Kappa$-deformed Poincarè algebra modifies the relativistic energy-momentum relation and the Lorentz transformations (on both space-time and four momentum coordinates), we were able to investigate the $\Kappa$-modification to a number of physical predictions of special relativity, including the Casimir effect, the relativistic Doppler effect, the Michelson Morley experiment and the Thomas Precession.

By comparing the above results with the experimental data, the following respective lower limits on $\Kappa$ were obtained: 0.3eV, 91eV, 6.2 keV and 210 MeV.

Low Energy Photon Interactions in Any Dimension

Adam Ritz, 1994.

The scattering of photons by photons is an interesting nonlinear phenomenon which, although forbidden by classical electromagnetism, is allowed by quantum electrodynamics in which the process proceeds via an interaction between photons and the vacuum of the Dirac field. The scattering cross-section for the lowest order process, $\ga +\ga \longrightarrow \ga+\ga$, is too small to be directly verified by experiment and thus, for theoretical purposes only, it is useful to consider the {\em effective Lagrangian\/} for the interaction which can be expressed in terms of Lorentz invariant functions of the electromagnetic field tensor $F(E,B)$.

For the low energy case the manifestly invariant effective interaction Lagrangian for 2, 4, and in general $N$-photon scattering has been determined in arbitrary dimensional space-time. This generalisation has revealed that the coefficients of the invariant functions in the Lagrangian have a structure based, not only on the dimensionality of space-time, but also on the generalised Riemann-Zeta function.

The Control of Chaos in the Rossler System

Stuart Corney, 1995.

Chaos is, by its very nature, unpredictable. It is this unpredictability which, until recently, had caused the presence of chaos in a real system to be so problematic. However it has been pointed out that by application of small, carefully chosen perturbations a chaotic system can be nudged onto one of an infinite number of unstable periodic orbits that exist on a strange attractor, thus controlling the system.

In this thesis the method for controlling chaos first demonstrated by Ott, Grebogi and Yorke is examined and applied to the Rossler system; a set of three dimensional, non-linear differential equations with three independent parameters. This analysis shows that the proposed method is sound, but only when certain assumptions are satisfied. The most notable of these assumptions is the ability to accurately linearise the dynamics of the system about a fixed point. When a fixed point is not linearisable the control method has no stabilising effect whatsoever.

Analysis of the Motion of a Bipolar Motor

Neville Jones, 1992.

Chaos theory is reshaping ideas about the behaviour of seemingly simple systems. An experimental device of a dipole free to rotate in a sinusoidally varying magnetic field has been utilized to study this field. The time evolution of the system is governed by a second-order nonlinear differential equation. Despite the ease with which the equation is derived, full analytical solutions are impossible.

The experimental system exhibits a rich variety of behaviour including periodic, period-doubled and chaotic. Support equipment permits detailed study of the dynamics from which numerous results have been derived. The transition points between the types of motion have been accurately determined, and a uniform band structure has been recognised where periodic solutions are separated by bands of chaotic behaviour. Universal features of chaotic systems have also been observed. The experimental results are confirmed by numerical simulation of the system.

The diversity of interesting behaviour the apparatus displays and its ability to produce replicable results make it ideal for use in the undergraduate experimental laboratory.

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