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Dynamics of Systems With Hamiltonian Monodromy
Salmon, Daniel
Salmon, Daniel
Abstract
A system is said to have monodromy if, when we carry the system around a closed circuit, it does not return to its initial state. The simplest example is the square-root function in the complex plane. A Hamiltonian system is said to have Hamiltonian monodromy if its fundamental action-angle loops do not return to their initial topological state at the end of a closed circuit. These changes in topology of angle loops carry through to other aspects of these systems, including the classical dynamics of families of trajectories, quantum spectra and even wavefunctions. This topological change in the evolution of a loop of classical trajectories has been observed experimentally for the rst time, using an apparatus consisting of a spherical pendulum subject to magnetic potentials and torques. Presented in this dissertation are the details of this experiment, as well as theoretical calculations on a novel system: a double welled Mexican-hat system with two monodromy points. This is part of a more general research program that is concerned with the Lagrangian torus bration of the phase spaces of integrable Hamiltonian systems. It is in this way the calculations on the double welled system are carried out. in this dissertation, static and dynamical manifestations of monodromy are shown to exist for this system. It has been shown previously that corresponding topological changes occur in wavefunctions of systems with monodromy. Here it is shown that results of quantum wavefunction monodomy carry over intuitively.
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2018-07-13
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Department
Physics
DOI
http://dx.doi.org/10.21220/s2-pr1x-h506