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Discrete reduction of mechanical systems and multisymplectic geometry of continuum mechanics


Pekarsky, Sergey (2000) Discrete reduction of mechanical systems and multisymplectic geometry of continuum mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1y6f-7w97.


This thesis develops discrete reduction techniques for mechanical systems defined on Lie groups and also presents multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. While the former synthesizes ideas of Euler-Poincaré and Lie-Poisson reduction for mechanical systems with the Veselov type discretization of such systems, the latter sets the stage for multisymplectic reduction and for further development of Veselov type multisymplectic discretizations. For systems defined on finite dimensional Lie groups G with Lagrangians L : TG → R that are G-invariant, the reduced discrete equations provide "reduced" numerical algorithms which manifestly preserve the underlying (symplectic) structure. The manifold G x G is used as an approximation of TG, and a discrete Lagrangian L : G x G → R is constructed in such a way that the G-invariance property is preserved. Reduction by G results in new "variational" principle for the reduced Lagrangian ℓ : G → R, and provides the discrete Euler-Poincaré (DEP) equations. The solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is also shown that the reduced Lagrangian ℓ : G → R defines a Poisson structure on (a subset) of one copy of the Lie group G. This structure governs the corresponding discrete reduced dynamics. The symplectic leaves of this structure become dynamically invariant manifolds which are manifestly preserved under the structure preserving discrete Euler-Poincaré algorithm. A variational multisymplectic formulation of non-relativistic continuum mechanics on general Riemannian manifolds is developed. Two main applications of our theory are considered — fluid dynamics and elasticity — each specified by a particular choice of the Lagrangian density. The non-relativistic character of the theory enables applications to such important cases as incompressible hydrodynamics and constrained director models of elastic rods and shells. These are applications of a general formalism developed here for treating non-relativistic first-order multisymplectic field theories with constraints. The results obtained in this thesis also set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms Combined with the ideas on discretizing systems with symmetries, this approach would results in so called multisymplectic integrators which preserve the discrete analogues of the conservation laws.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Control and Dynamical Systems
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):
  • Marsden, Jerrold E.
Thesis Committee:
  • Unknown, Unknown
Defense Date:18 May 2000
Record Number:CaltechTHESIS:10062010-115304008
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6113
Deposited By: Benjamin Perez
Deposited On:07 Oct 2010 16:28
Last Modified:16 Apr 2021 22:29

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