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Geometric Descriptions of Couplings in Fluids and Circuits

Citation

Jacobs, Henry Ochi (2012) Geometric Descriptions of Couplings in Fluids and Circuits. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/AZXE-PH33. https://resolver.caltech.edu/CaltechTHESIS:04302012-142612208

Abstract

Geometric mechanics is often commended for its breadth (e.g., fluids, circuits, controls) and depth (e.g., identification of stability criteria, controllability criteria, conservation laws). However, on the interface between disciplines it is commonplace for the analysis previously done on each discipline in isolation to break down. For example, when a solid is immersed in a fluid, the particle relabeling symmetry is broken because particles in the fluid behave differently from particles in the solid. This breaks conservation laws, and even changes the configuration manifolds. A second example is that of the interconnection of circuits. It has been verified that LC-circuits satisfy a variational principle. However, when two circuits are soldered together this variational principle must transform to accommodate the interconnection.

Motivated by these difficulties, this thesis analyzes the following couplings: fluid-particle, fluid-structure, and circuit-circuit. For the case of fluid-particle interactions we understand the system as a Lagrangian system evolving on a Lagrange-Poincare bundle. We leverage this interpretation to propose a class of particle methods by "ignoring" the vertical Lagrange-Poincare equation. In a similar vein, we can analyze fluids interacting with a rigid body. We then generalize this analysis to view fluid-structure problems as Lagrangian systems on a Lie algebroid. The simplicity of the reduction process for Lie algebroids allows us to propose a mechanism in which swimming corresponds to a limit-cycle in a reduced Lie algebroid. In the final section we change gears and understand non-energetic interconnection as Dirac structures. In particular we find that any (linear) non-energetic interconnection is equivalent to some Dirac structure. We then explore what this insight has to say about variational principles, using interconnection of LC-circuits as a guiding example.

Item Type: Thesis (Dissertation (Ph.D.)) geometric mechanics, symplectic geometry, Lie groupoids, Lie algebroids, Lagrange Poincare, Dirac structures, Lagrangian systems, Hamiltonian systems, Poisson structures California Institute of Technology Engineering and Applied Science Control and Dynamical Systems Public (worldwide access) Desbrun, Mathieu Desbrun, Mathieu (chair)Murray, Richard M.Owhadi, HoumanKanso, Eva 20 April 2012 CaltechTHESIS:04302012-142612208 https://resolver.caltech.edu/CaltechTHESIS:04302012-142612208 10.7907/AZXE-PH33 No commercial reproduction, distribution, display or performance rights in this work are provided. 6991 CaltechTHESIS Henry Jacobs 14 May 2012 19:00 03 Oct 2019 23:55

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