Discrete Euler-Poincar<metaTags></metaTags>#x27;{e} and Lie-Poisson Equations
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## Discrete Euler-Poincar window.jscholApp_initialPageData = {"added":"2018-02-15","advisors":null,"altmetrics_ok":true,"appearsIn":[{"id":"ucdavismath_faculty","name":"Faculty"}],"attrs":{"doi":"10.1088/0951-7715/12/6/314","abstract":"In this paper, discrete analogues of Euler-Poincar\\'{e} and Lie-Poisson reduction\n theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians\n $L:TG \\to {\\mathbb R}$ that are $G$-invariant. These discrete equations provide reduced''\n numerical algorithms which manifestly preserve the symplectic structure. The manifold $G\n \\times G$ is used as an approximation of $TG$, and a discrete Langragian ${\\mathbb L}:G\n \\times G \\to {\\mathbb R}$ is construced in such a way that the $G$-invariance property is\n preserved. Reduction by $G$ results in new variational'' principle for the reduced\n Lagrangian $\\ell:G \\to {\\mathbb R}$, and provides the discrete Euler-Poincar\\'{e} (DEP)\n equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations\n developed in \\cite{MPS,WM} which are naturally symplectic-momentum algorithms. Furthermore,\n the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP)\n algorithm. It is shown that when $G=\\text{SO} (n)$, the DEP and DLP algorithms for a\n particular choice of the discrete Lagrangian ${\\mathbb L}$ are equivalent to the\n Moser-Veselov scheme for the generalized rigid body. %As an application, a reduced\n symplectic integrator for two dimensional %hydrodynamics is constructed using the SU$(n)$\n approximation to the volume %preserving diffeomorphism group of ${\\mathbb T}^2$.","subjects":["math.NA","math.SG"],"local_ids":[{"id":"math/9909099","type":"arXiv"},{"id":"UC Davis Math 1999-32","type":"other"}],"submitter":"help@escholarship.org","pub_status":"externalPub","pub_web_loc":["https://arxiv.org/pdf/math/9909099.pdf"],"is_peer_reviewed":true},"authors":[{"name":"Marsden, Jerrold E.","fname":"Jerrold E.","lname":"Marsden"},{"name":"Pekarsky, Sergey","fname":"Sergey","lname":"Pekarsky"},{"name":"Shkoller, Steve","fname":"Steve","lname":"Shkoller"}],"citation":{"id":"qt6m51m12s","type":"article","title":"Discrete Euler-Poincar\\'{e} and 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Lie-Poisson Equations

• Author(s): Marsden, Jerrold E.
• Pekarsky, Sergey
• Shkoller, Steve
• et al.

## Published Web Location

https://arxiv.org/pdf/math/9909099.pdf
No data is associated with this publication.
Abstract

In this paper, discrete analogues of Euler-Poincar {e} and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete equations provide reduced'' numerical algorithms which manifestly preserve the symplectic structure. The manifold $G \times G$ is used as an approximation of $TG$, and a discrete Langragian ${\mathbb L}:G \times G \to {\mathbb R}$ is construced in such a way that the $G$-invariance property is preserved. Reduction by $G$ results in new variational'' principle for the reduced Lagrangian $\ell:G \to {\mathbb R}$, and provides the discrete Euler-Poincar {e} (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when $G=\text{SO} (n)$, the DEP and DLP algorithms for a particular choice of the discrete Lagrangian ${\mathbb L}$ are equivalent to the Moser-Veselov scheme for the generalized rigid body. %As an application, a reduced symplectic integrator for two dimensional %hydrodynamics is constructed using the SU$(n)$ approximation to the volume %preserving diffeomorphism group of ${\mathbb T}^2$.

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