Point Vortices: Finding Periodic Orbits and their Topological Classification.
Abstract: 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 ... read moremodeled 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.
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Physics.
Advisor: Bruce Boghosian.
Committee: Roger Tobin, Krzysztof Sliwa, Timothy Atherton, and John Gibson.
Keywords: Physics, and Mathematics.read less
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