Abstract and Physical Effects of Curvature on Dynamics of Extended Body Systems
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Day, Brian
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Abstract
The presence of intrinsic curvature of an ambient space influences the dynamics of point particles moving through it as typically considered in applications of differential geometry in physical contexts, such as general relativity. We aim to utilize the mathematics of differential geometry to instead consider the collective curvature effects on extended body systems in some generic curved space. To this end we develop a mathematical framework which serves as the foundation of a general dynamics solver numerical toolkit in which users can simulate the dynamics of discrete extended body systems in generic curved spaces. Through analyzing the dynamics of such extended body systems we recognized a relationship between deformation of the body during its dynamics as a result of the ambient curvature. This led us to expand our mathematical model of extended bodies to include deformable bodies. We find that such deformable bodies can generate collective motion via deforming their body even in a ambient space lacking curvature. This is due to the presence of an abstract notion of curvature defined on the configuration space of the system via considering the system as being described by a mathematical object known as a fiber bundle. This revelation allows us to discuss the dynamics of such deformable control systems using the ideas of geometric mechanics. In particular, we consider recasting our system in a geometric mechanics framework to address the question of determining optimal controls of how to deform the system so as to minimize some cost function. This is based on considering the optimization problem as a variational problem whose solutions correspond to optimal controls of the system. We develop this variational approach into a numerical toolkit acting as the foundation of a more general purpose optimization toolkit for deformable control systems described by fibers bundles.
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2023-12-07
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Dissertation