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USING LIE GROUP INTEGRATORS TO SOLVE TWO DIMENSIONAL VARIATIONAL PROBLEMS WITH SYMMETRY

Zadra, Michele and Mansfield, Elizabeth L. (2019) USING LIE GROUP INTEGRATORS TO SOLVE TWO DIMENSIONAL VARIATIONAL PROBLEMS WITH SYMMETRY. Technical report. American Institute of Mathematical Sciences (Submitted)

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Abstract

The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. Unlike in the 1D case, where Noether’s laws give first integrals of the Euler–Lagrange equations, in higher dimensional problems the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use a moving frame to solve, numerically, a two dimensional (2D) variational problem, invariant under a projective action of SL(2). In order to find a solution to the variational problem, we may solve a related 2D system of linear, first order, coupled ODEs for the moving frame, evolving on SL(2). We demonstrate that Lie group integrators [12] may be used in this context, by showing that such systems are also numerically compatible, up to order 5, that is, the result is independent of the order of integration. This compatibility is a testament to the level of geometry built into the Lie group integrators.

Item Type: Monograph (Technical report)
Uncontrolled keywords: calculus of variations, Lie groups
Subjects: Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus
Q Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User: Elizabeth L Mansfield
Date Deposited: 28 Mar 2019 15:06 UTC
Last Modified: 03 Jun 2019 09:37 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/73255 (The current URI for this page, for reference purposes)
Mansfield, Elizabeth L.: https://orcid.org/0000-0002-6778-2241
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