The motion of point vortices constitutes an especially simple class of solutions to Euler's equation for two dimensional, inviscid, incompressible, and irrotational fluids. In addition to their intrinsic mathematical importance, these solutions are also physically relevant. Rotating superfluid helium can support rectilinear quantized line vortices, which in certain regimes are accurately modeled by point vortices. Depending on the number of vortices, it is possible to have either regular integrable motion or chaotic motion. Thus, the point vortex model is one of the simplest and most tractable fluid models which exhibits some of the attributes of weak turbulence.
The primary aim of this work is to find and classify periodic orbits, a special class of solutions to the point vortex problem. To achieve this goal, we introduce a number of algorithms: Lie transforms which ensure that the equations of motion are accurately solved; constrained optimization which reduces close return orbits to true periodic orbits; object-oriented representations of the braid group which allow for the topological comparison of periodic orbits. By applying these ideas, we accumulate a large data set of periodic orbits and their associated attributes. To render this set tractable, we introduce a topological classification scheme based on a natural decomposition of mapping classes. Finally, we consider some of the intriguing patterns which emerge in the distribution of periodic orbits in phase space. Perhaps the most enduring theme which arises from this investigation is the interplay between topology and geometry. The topological properties of a periodic orbit will often force it to have certain geometric properties.
|Advisor:||Boghosian, Bruce M.|
|Commitee:||Atherton, Timothy, Gibson, John, Sliwa, Krzysztof, Tobin, Roger|
|School Location:||United States -- Massachusetts|
|Source:||DAI-B 74/03(E), Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics, Physics, Theoretical physics|
|Keywords:||Algorithm, Braids, Hamiltonian, Periodic orbits, Point vortex, Topology|
Copyright in each Dissertation and Thesis is retained by the author. All Rights Reserved
The supplemental file or files you are about to download were provided to ProQuest by the author as part of a
dissertation or thesis. The supplemental files are provided "AS IS" without warranty. ProQuest is not responsible for the
content, format or impact on the supplemental file(s) on our system. in some cases, the file type may be unknown or
may be a .exe file. We recommend caution as you open such files.
Copyright of the original materials contained in the supplemental file is retained by the author and your access to the
supplemental files is subject to the ProQuest Terms and Conditions of use.
Depending on the size of the file(s) you are downloading, the system may take some time to download them. Please be