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From Differential to Difference: The Variational Bicomplex and Invariant Noether's Theorems.

Peng, Linyu. (2013) From Differential to Difference: The Variational Bicomplex and Invariant Noether's Theorems. Doctoral thesis, University of Surrey (United Kingdom)..

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A solution of a differential system can be interpreted as a maximal submanifold determined by the Cartan distribution on jet bundle. By using an exact couple, we obtain the spectral sequence associated with the bigraded structure or equivalently the differential variational bicomplex on the jet bundle, that tremendously simplifies Vinogradov’s approach. The spectral sequence consists of leaves of cohomology groups, which are closely related to conservation laws. Similarly to the jet bundle structure, we construct a space for difference systems, the prolongation bundle, on which the difference variational bicomplex is built accordingly. It allows us to describe discrete mechanics globally and is applied to find conservation laws for (difference) multisymplectic systems. The exact couple methodology is applied to the corresponding (difference) bigraded structure and shows a coherence between the differential and difference cases. When the difference system contributes to an empty constraint, we prove the exactness of an amended (difference) bicomplex. This thesis also focuses on the applications of the moving frame method to Noether’s theorems. We generalize group actions on smooth manifolds to the prolongation bundle, which helps us to establish a discrete counterpart of a moving frame. This is used to illustrate a discrete version of Noether’s first theorem by taking a relevant variational symmetry group action and its Adjoint action into consideration. We also consider Noether’s second theorem dealing with invariant variational problems whose symmetry characteristics are determined by some smooth functions. The Adjoint action is used to achieve the differential relationships among the invariant Euler-Lagrange equations. When symmetry characteristics depend on constrained smooth functions, conservation laws with respect to the related Euler-Lagrange equations can be constructed. The method is illustrated by some physical examples possessing (local) SU(n) gauge symmetries.

Item Type: Thesis (Doctoral)
Divisions : Theses
Authors : Peng, Linyu.
Date : 2013
Additional Information : Thesis (Ph.D.)--University of Surrey (United Kingdom), 2013.
Depositing User : EPrints Services
Date Deposited : 06 May 2020 14:23
Last Modified : 06 May 2020 14:28

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