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Moving Frames and Noether’s Finite Difference Conservation Laws I

Mansfield, Elizabeth L., Rojo-Echeburua, Ana, Hydon, Peter E., Peng, Linyu (2019) Moving Frames and Noether’s Finite Difference Conservation Laws I. Transactions of Mathematics and its Applications, 3 (1). pp. 1-47. E-ISSN 2398-4945. (doi:10.1093/imatrm/tnz004) (KAR id:74247)

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We consider the calculation of Euler--Lagrange systems of ordinary difference equations, including the difference Noether's Theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler-Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether's conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler--Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant.

Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded \textit{a priori}. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts.

We show the calculations for a discretization of the Lagrangian for Euler's elastica, and compare our discrete solution to that of its smooth continuum limit.

Item Type: Article
DOI/Identification number: 10.1093/imatrm/tnz004
Uncontrolled keywords: Noether's Theorem, Finite Difference, Discrete Moving Frames
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Elizabeth Mansfield
Date Deposited: 04 Jun 2019 13:46 UTC
Last Modified: 16 Feb 2021 14:04 UTC
Resource URI: (The current URI for this page, for reference purposes)
Mansfield, Elizabeth L.:
Hydon, Peter E.:
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