Citation
Jacobs, Henry Ochi (2012) Geometric descriptions of couplings in fluids and circuits. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:04302012142612208
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 LCcircuits 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: fluidparticle, fluidstructure, and circuitcircuit. For the case of fluidparticle interactions we understand the system as a Lagrangian system evolving on a LagrangePoincare bundle. We leverage this interpretation to propose a class of particle methods by ``ignoring'' the vertical LagrangePoincare equation. In a similar vein, we can analyze fluids interacting with a rigid body. We then generalize this analysis to view fluidstructure 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 limitcycle in a reduced Lie algebroid. In the final section we change gears and understand nonenergetic interconnection as Dirac structures. In particular we find that any (linear) nonenergetic interconnection is equivalent to some Dirac structure. We then explore what this insight has to say about variational principles, using interconnection of LCcircuits as a guiding example.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  geometric mechanics, symplectic geometry, Lie groupoids, Lie algebroids, Lagrange Poincare, Dirac structures, Lagrangian systems, Hamiltonian systems, Poisson structures 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Control and Dynamical Systems 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  20 April 2012 
Record Number:  CaltechTHESIS:04302012142612208 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:04302012142612208 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6991 
Collection:  CaltechTHESIS 
Deposited By:  Henry Jacobs 
Deposited On:  14 May 2012 19:00 
Last Modified:  22 Aug 2016 21:23 
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