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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). E-ISSN 2398-4945. (doi:10.1093/imatrm/tnz004)

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https://doi.org/10.1093/imatrm/tnz004

Abstract

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 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: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Elizabeth Mansfield
Date Deposited: 04 Jun 2019 13:46 UTC
Last Modified: 11 Oct 2019 13:23 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/74247 (The current URI for this page, for reference purposes)
Mansfield, Elizabeth L.: https://orcid.org/0000-0002-6778-2241
Hydon, Peter E.: https://orcid.org/0000-0002-3732-4813
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