# Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

Zadra, Michele, Mansfield, Elizabeth L. (2019) Using Lie group integrators to solve two and higher dimensional variational problems with symmetry. Journal of Computational Dynamics, 6 (2). pp. 485-511. ISSN 2158-2491. (doi:10.3934/jcd.2019025) (KAR id:73255)

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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. In the one dimensional case, 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 the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of $$SU(2)$$. We then consider variational problems for evolving curves which are invariant under the projective action of $$SL(2)$$ and finally the standard affine action of $$SE(2)$$.

Item Type: Article 10.3934/jcd.2019025 calculus of variations, Moving frames, symmetries, Lie groups, Lie group integrators, PDEs Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Elizabeth Mansfield 28 Mar 2019 15:06 UTC 16 Feb 2021 14:03 UTC https://kar.kent.ac.uk/id/eprint/73255 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-6778-2241