High-order symplectic Lie group methods on \(SO(n)\) using the polar decomposition
type: Item
instance of: scholarly article
title: High-order symplectic Lie group methods on \(SO(n)\) using the polar decomposition
author: Xuefeng Shen
author: Hoa Tran
author: Melvin Leok
published in: Journal of Computational Dynamics
publication date: 2022-11-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/2201.10768
review text: The authors construct a variational integrator of arbitrarily high-order on the special orthogonal group \(\mathrm{SO}(n)\) using the polar decomposition and the constrained Galerkin method. \(\mathrm{SO}(n)\) is embedded naturally in the space \(\mathrm{GL}_{+}(n)=\{A\in \mathbb{R}^{n\times n} | \mathrm{det}(A) > 0\}\), an open subset of \(\mathbb{R}^{n\times n}\). It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. The Lie algebra of \(\mathrm{SO}(n)\) is the set Skew(\(n\)) \(= \{\Omega \in \mathbb{R}^{n\times n} | \Omega^{T} = -\Omega\}\), with the matrix commutator as the Lie bracket. The authors introduce the polar decomposition and the retraction on \(\mathrm{SO}(n)\). The polar decomposition of \(A\) is provided according to the following: Given \(A\in \mathrm{GL}(n)\), this decomposes uniquely as \(A\rightarrow UP\), \(U\in \mathrm{O}(n)\), \(P\in \mathrm{Sym}^{+}(n)\), where \(\mathrm{Sym}^{+}(n)\) is the set of \(n \times n\) symmetric positive-definite matrices. The authors also mention the Lagrangian variational integrators on the rotation group \(\mathrm{SO}(n)\) in details. For a system with rotational symmetry, the authors obtain a simpler integrator using Lie-Poisson reduction on the Hamiltonian side. Further, they construct a reduced Lie-Poisson integrator and the resulting algorithms can naturally be implemented by fixed-point iteration. The authors validate the proposed methods by numerical simulations on \(\mathrm{SO}(3)\) showing they are comparable to variational Runge-Kutta-Munthe-Kaas methods in terms of computational efficiency. They provide the following methods for the comparison tests: second-order Gauss-Legendre method, third-order Runge-Kutta method, fourth-order Gauss-Legendre method and sixth-order Gauss-Legendre method. However, these proposed methods preserve the Lie group structure much more accurately and exhibit better near energy preservat…
reviewed by: Mohammad Sajid
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 65P10
Mathematics Subject Classification ID: 65J05
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 70H25
Mathematics Subject Classification ID: 47J26
Mathematics Subject Classification ID: 58J20
zbMATH DE Number: 7613237
zbMATH Keywords: geometric numerical integration
zbMATH Keywords: symplectic integration
zbMATH Keywords: variational integrator
zbMATH Keywords: Lie group integrator
zbMATH Keywords: rotation group
zbMATH Keywords: polar decomposition
MaRDI profile type: Publication
OpenAlex ID: W4214743283
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DOI: 10.3934/JCD.2022003
description: scientific article; zbMATH DE number 7613237