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Geometric Discretization of Lagrangian Mechanics and Field Theories


Stern, Ari Joshua (2009) Geometric Discretization of Lagrangian Mechanics and Field Theories. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K943-VJ44.


This thesis presents a unified framework for geometric discretization of highly oscillatory mechanics and classical field theories, based on Lagrangian variational principles and discrete differential forms. For highly oscillatory problems in mechanics, we present a variational approach to two families of geometric numerical integrators: implicit-explicit (IMEX) and trigonometric methods. Next, we show how discrete differential forms in spacetime can be used to derive a structure-preserving discretization of Maxwell's equations, with applications to computational electromagnetics. Finally, we sketch out some future directions in discrete gauge theory, providing foundations based on fiber bundles and Lie groupoids, as well as discussing applications to discrete Riemannian geometry and numerical general relativity.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:discrete differential geometry; discrete Hodge star; Fermi-Pasta-Ulam problem; frame bundle; geometric mechanics; mixed finite elements; multisymplectic; Regge calculus; symplectic; variational integrators; variational methods
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied And Computational Mathematics
Awards:The W.P. Carey and Co., Inc., Prize in Applied Mathematics, 2009.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Marsden, Jerrold E. (advisor)
  • Desbrun, Mathieu (advisor)
Thesis Committee:
  • Marsden, Jerrold E. (chair)
  • Owhadi, Houman
  • Desbrun, Mathieu
  • Schroeder, Peter
  • Hou, Thomas Y.
Defense Date:5 December 2008
Record Number:CaltechETD:etd-12312008-173851
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:5173
Deposited By: Imported from ETD-db
Deposited On:23 Jan 2009
Last Modified:26 Nov 2019 19:14

Thesis Files

PDF (Stern_PhD_Thesis_Single-Sided.pdf) - Final Version
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PDF (Stern_PhD_Thesis_Double-Sided.pdf) - Final Version
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