
Dynamics in the Hopf bundle and the
geometric phase

 

A Hopf bundle framework is constructed within C^n, in terms of which
general paths on C^n\{0} are viewed and analyzed. The resulting hierarchy of spaces is addressed both theoretically and numerically, and the consequences for numerics and applications are investigated through a wide range of numerical experiments.
The geometric reframing of C^n in this way  in terms of an intrinsic fibre bundle 
allows for the introduction of bundletheoretic quantities in a general dynamical setting. The roles of the various structural elements of the bundle are explored, including horizontal and vertical subspaces, parallel translation and connections. These concepts lead naturally to the association of a unique geometric phase with each path on C^n\{0}. This phase quantity is interpreted as a measure of the spinning in the S^1 fibre of the Hopf bundle induced by paths on C^n\{0}, relative to a given connection, and is shown to be an important quantity.
The implications of adopting this bundle viewpoint are investigated in
two specific contexts. The first is the case of the lowestdimensional
Hopf bundle, S^1 > S^3 > S^2. Here the quaternionic matrices are
used to develop a simplified, geometrically intuitive formulation of the bundle structure, and a reduced expression for the phase is used to compute numerical phase results in three example systems.
The second is the case where paths in C^n\{0} are generated by solutions to a particular class of parameterdependent firstorder ODEs. This establishes a direct link between the dynamical characteristics of such systems and the underlying bundle geometry. A variety of systems are examined and numerical phase results compiled. The numerics reveal an important correlation between the spectral properties of the pathgenerating ODEs and the resultant geometric phase change values. The details of this observed link are
recorded in a conjecture.

 

R. Way.
Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems.
PhD Thesis, University of Surrey (2009)


