December 1998, The University of Texas at Austin. Supervisor: Philip J. Morrison
We classify Lie-Poisson brackets that are formed from Lie algebra extensions. The problem is relevant because many physical systems owe their Hamiltonian structure to such brackets. A classification involves reducing all brackets to a set of normal forms, independent under coordinate transformations, and is achieved with the techniques of Lie algebra cohomology. For extensions of order less than five, we find that the number of normal forms is small and they involve no free parameters. A special extension, known as the Leibniz extension, is shown to be the unique ``maximal'' extension.
We derive a general method of finding Casimir invariants of Lie--Poisson bracket extensions. The Casimir invariants of all brackets of order less than five are explicitly computed, using the concept of coextension. We obtain the Casimir invariants of Leibniz extensions of arbitrary order. We also offer some physical insight into the nature of the Casimir invariants of compressible reduced magnetohydrodynamics.
We make use of the methods developed to study the stability of extensions for given classes of Hamiltonians. This helps to elucidate the distinction between semidirect extensions and those involving cocycles. For compressible reduced magnetohydrodynamics, we find the cocycle has a destabilizing effect on the steady-state solutions.
There is a PDF file of the thesis available. Also, see the publications section. Available as Institute for Fusion Studies Report #847.
August 1995, The University of Texas at Austin. Supervisor: Wendell Horton
The Partial Differential Equations (PDE's) for Rayleigh-Benard convection of a fluid between two plates with free boundary conditions are turned into an infinite system of coupled Ordinary Differential Equations (ODE's) by expansion in Fourier modes. Shear flow and variable phase between modes are allowed. A general method is presented to make finite truncations of this system that preserve the invariants of the full PDE's in the ideal limit. These truncations also have the property that they have no unbounded solutions and provide a description of the heat flux that has the correct limiting behaviour in a steady-state. A particular truncation (containing 7 modes) is selected and is compared to a previous model, the 6-ODE model of Howard and Krishnamurti (1986). Numerical calculations are presented to compare the two truncations and study the effects of shear flow on heat transport.
There is PDF file of the thesis available. Also, see the publications section. A short (2 pages) related article, Energy Conserving Truncations in Thermal Convection, was published in the IFS newsletter, XI, 1. Also see the preprints section.