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Experiments And Theory On Dynamical Hamiltononian Monodromy
Nerem, Matthew Perry
Nerem, Matthew Perry
Abstract
In classical mechanics, one of the advanced topics is the study of action and angle variables. These variables are quite abstract, but very powerful tools for describing classical motion. If a system has a full set of conservation laws, and if the motion of the system is bounded, then the motion can be described as flow on a torus. Action variables are functions of the conservation laws that identify the torus on which the motion lies, while angle variables tell the location of the system on that torus. In certain cases, the functional relationship between the conservation laws and the action variables has singular points, and these singular points can cause the action variables to become multivalued functions. These multivalued action variables have surprising topological and dynamical consequences. Which become visible when we follow the evolution of a family of trajectories in a system having multivalued action variables under the influence externally applied forces: a loop of particles evolves smoothly to a topologically different loop. We call this phenomenon Dynamical Hamiltonian Monodromy. This thesis reports the first experimental observation of Dynamical Hamiltonian Monodromy. A spherical pendulum controlled by magnetic forces was used to generate a topological change in a family of pendulum trajectories. It then explores application of this theory to an optical system and determines the index of refraction of a light pipe that similarly will produce the topological change connected with Hamiltonian Monodromy.
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2020-01-01
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Physics
DOI
http://dx.doi.org/10.21220/s2-tgn5-0t18