Discrete Euler-Poincar<metaTags></metaTags>#x27;{e} and Lie-Poisson Equations
Open Access Publications from the University of California

## 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 Lie-Poisson Equations","URL":"http://escholarship.org/uc/item/6m51m12s","issued":{"raw":["1999-9-17"]},"doi":"10.1088/0951-7715/12/6/314","issn":null,"author":[{"literal":"Marsden, Jerrold E.","given":"Jerrold E.","family":"Marsden"},{"literal":"Pekarsky, Sergey","given":"Sergey","family":"Pekarsky"},{"literal":"Shkoller, Steve","given":"Steve","family":"Shkoller"}]},"content_html":false,"content_key":"ffd64d72fcd47d94e128309076ecd41b","content_type":null,"data_digest":"my2XNPRHo1gSMJqzDdXzdA==","editors":null,"genre":"article","id":"6m51m12s","index_digest":"RnNEK6pIyA/Rk5JIlnBcjg==","last_indexed":"2019-02-02 20:30:05 +0000","oa_policy":null,"ordering_in_sect":null,"pdf_url":null,"published":"1999-09-17","rights":null,"sidebar":[{"id":1,"kind":"RecentArticles","attrs":{"items":[{"id":"qt01c0p4jn","title":"Editors note: Spontaneous <i>SU</i><sub>2</sub>(C) symmetry breaking in the ground states of quantum spin chain (Journal of Mathematical Physics (2018) 59 (111701) DOI: 10.1063/1.5078597)","authors":[{"name":"Nachtergaele, B","email":"bnachtergaele@ucdavis.edu","fname":"B","lname":"Nachtergaele","ORCID_id":"0000-0002-7835-3776"}],"genre":"article","author_hide":null},{"id":"qt30d0p416","title":"Unified theory for finite Markov chains","authors":[{"name":"Rhodes, John","fname":"John","lname":"Rhodes"},{"name":"Schilling, Anne","email":"aschilling@ucdavis.edu","fname":"Anne","lname":"Schilling","ORCID_id":"0000-0002-2601-7340"}],"genre":"article","author_hide":null},{"id":"qt31k9s95q","title":"Mechanosensitive Adhesion Explains Stepping Motility in Amoeboid Cells","authors":[{"name":"Copos, CA","fname":"CA","lname":"Copos"},{"name":"Walcott, S","fname":"S","lname":"Walcott"},{"name":"del \u00C1lamo, JC","fname":"JC","lname":"del \u00C1lamo"},{"name":"Bastounis, E","fname":"E","lname":"Bastounis"},{"name":"Mogilner, A","fname":"A","lname":"Mogilner"},{"name":"Guy, RD","email":"rdguy@ucdavis.edu","fname":"RD","lname":"Guy"}],"genre":"article","author_hide":null},{"id":"qt4f58m5rx","title":"The support of integer optimal solutions","authors":[{"name":"Aliev, I","fname":"I","lname":"Aliev"},{"name":"De Loera, JA","email":"jadeloera@ucdavis.edu","fname":"JA","lname":"De Loera"},{"name":"Eisenbrand, F","fname":"F","lname":"Eisenbrand"},{"name":"Oertel, T","fname":"T","lname":"Oertel"},{"name":"Weismantel, R","fname":"R","lname":"Weismantel"}],"genre":"article","author_hide":null},{"id":"qt5cp2c1bd","title":"Common genetic variants influence human subcortical brain structures","authors":[{"name":"Hibar, DP","fname":"DP","lname":"Hibar"},{"name":"Stein, JL","fname":"JL","lname":"Stein"},{"name":"Renteria, ME","fname":"ME","lname":"Renteria"},{"name":"Arias-Vasquez, A","fname":"A","lname":"Arias-Vasquez"},{"name":"Desrivi\u00E8res, S","fname":"S","lname":"Desrivi\u00E8res"},{"name":"Jahanshad, N","fname":"N","lname":"Jahanshad"},{"name":"Toro, R","fname":"R","lname":"Toro"},{"name":"Wittfeld, K","fname":"K","lname":"Wittfeld"},{"name":"Abramovic, L","fname":"L","lname":"Abramovic"},{"name":"Andersson, M","fname":"M","lname":"Andersson"},{"name":"Aribisala, BS","fname":"BS","lname":"Aribisala"},{"name":"Armstrong, NJ","fname":"NJ","lname":"Armstrong"},{"name":"Bernard, M","fname":"M","lname":"Bernard"},{"name":"Bohlken, MM","fname":"MM","lname":"Bohlken"},{"name":"Boks, MP","fname":"MP","lname":"Boks"},{"name":"Bralten, J","fname":"J","lname":"Bralten"},{"name":"Brown, AA","fname":"AA","lname":"Brown"},{"name":"Mallar Chakravarty, M","fname":"M","lname":"Mallar Chakravarty"},{"name":"Chen, Q","fname":"Q","lname":"Chen"},{"name":"Ching, CRK","fname":"CRK","lname":"Ching"},{"name":"Cuellar-Partida, G","fname":"G","lname":"Cuellar-Partida"},{"name":"Den Braber, A","fname":"A","lname":"Den Braber"},{"name":"Giddaluru, S","fname":"S","lname":"Giddaluru"},{"name":"Goldman, AL","fname":"AL","lname":"Goldman"},{"name":"Grimm, O","fname":"O","lname":"Grimm"},{"name":"Guadalupe, T","fname":"T","lname":"Guadalupe"},{"name":"Hass, J","email":"jhass@ucdavis.edu","fname":"J","lname":"Hass"},{"name":"Woldehawariat, G","fname":"G","lname":"Woldehawariat"},{"name":"Holmes, AJ","fname":"AJ","lname":"Holmes"},{"name":"Hoogman, M","fname":"M","lname":"Hoogman"},{"name":"Janowitz, D","fname":"D","lname":"Janowitz"},{"name":"Jia, T","fname":"T","lname":"Jia"},{"name":"Kim, S","fname":"S","lname":"Kim"},{"name":"Klein, M","fname":"M","lname":"Klein"},{"name":"Kraemer, B","fname":"B","lname":"Kraemer"},{"name":"Lee, PH","fname":"PH","lname":"Lee"},{"name":"Olde Loohuis, LM","fname":"LM","lname":"Olde Loohuis"},{"name":"Luciano, M","fname":"M","lname":"Luciano"},{"name":"MacAre, C","fname":"C","lname":"MacAre"},{"name":"Mather, KA","fname":"KA","lname":"Mather"},{"name":"Mattheisen, M","fname":"M","lname":"Mattheisen"},{"name":"Milaneschi, Y","fname":"Y","lname":"Milaneschi"},{"name":"Nho, K","fname":"K","lname":"Nho"},{"name":"Papmeyer, M","fname":"M","lname":"Papmeyer"},{"name":"Ramasamy, A","fname":"A","lname":"Ramasamy"},{"name":"Risacher, SL","fname":"SL","lname":"Risacher"},{"name":"Roiz-Santia\u00F1ez, R","fname":"R","lname":"Roiz-Santia\u00F1ez"},{"name":"Rose, EJ","fname":"EJ","lname":"Rose"},{"name":"Salami, A","fname":"A","lname":"Salami"},{"name":"S\u00E4mann, PG","fname":"PG","lname":"S\u00E4mann"},{"name":"Schmaal, L","fname":"L","lname":"Schmaal"},{"name":"Schork, AJ","fname":"AJ","lname":"Schork"},{"name":"Shin, J","fname":"J","lname":"Shin"},{"name":"Strike, LT","fname":"LT","lname":"Strike"},{"name":"Teumer, A","fname":"A","lname":"Teumer"},{"name":"Van Donkelaar, MMJ","fname":"MMJ","lname":"Van Donkelaar"},{"name":"Van Eijk, KR","fname":"KR","lname":"Van Eijk"},{"name":"Walters, RK","fname":"RK","lname":"Walters"},{"name":"Westlye, LT","fname":"LT","lname":"Westlye"},{"name":"Whelan, CD","fname":"CD","lname":"Whelan"}],"genre":"article","author_hide":null}],"title":"Related Items"}}],"source":"subi","status":"published","submitted":"2017-06-13","title":"Discrete Euler-Poincar\\'{e} and Lie-Poisson Equations","unit":{"id":"ucdavismath_faculty","name":"Faculty","type":"series","status":"active"},"usage":[{"month":"2017-06","hits":0,"downloads":0},{"month":"2018-02","hits":1,"downloads":0},{"month":"2019-04","hits":1,"downloads":0}],"unit_attrs":{"about":"About Faculty: TODO","nav_bar":[]},"header":{"campusID":"ucd","campusName":"UC Davis","ancestorID":"ucdavismath","ancestorName":"Department of Mathematics","campuses":[{"id":"","name":"eScholarship at..."},{"id":"ucb","name":"UC Berkeley"},{"id":"ucd","name":"UC Davis"},{"id":"uci","name":"UC Irvine"},{"id":"ucla","name":"UCLA"},{"id":"ucm","name":"UC Merced"},{"id":"ucr","name":"UC Riverside"},{"id":"ucsd","name":"UC San Diego"},{"id":"ucsf","name":"UCSF"},{"id":"ucsb","name":"UC Santa Barbara"},{"id":"ucsc","name":"UC Santa Cruz"},{"id":"ucop","name":"UC Office of the President"},{"id":"lbnl","name":"Lawrence Berkeley National Laboratory"},{"id":"anrcs","name":"UC Agriculture & Natural Resources"}],"logo":{"url":"/assets/f201c79605f721b5015a79efed6489014d1aa8d631abeb749a4751532c7d25ab","width":505,"height":58,"is_banner":true},"directSubmit":null,"directSubmitURL":null,"nav_bar":[{"id":-9999,"type":"fixed_page","name":"Unit Home","url":"/uc/ucdavismath"},{"id":1,"name":"About","type":"folder","sub_nav":[{"id":2,"name":"About Us","slug":"aboutUs","type":"page","url":"/uc/ucdavismath/aboutUs"}]},{"id":3,"name":"Policies for Submissions","slug":"policyStatement","type":"page","url":"/uc/ucdavismath/policyStatement"},{"id":4,"name":"other pages","type":"folder","hidden":true,"sub_nav":[]}],"social":{"facebook":null,"twitter":null,"rss":"/rss/unit/ucdavismath_faculty"},"breadcrumb":[{"name":"eScholarship","id":"root","url":"/"},{"name":"UC Davis","id":"ucd","url":"/uc/ucd"},{"name":"Department of Mathematics","id":"ucdavismath","url":"/uc/ucdavismath"},{"name":"Faculty","id":"ucdavismath_faculty","url":"/uc/ucdavismath_faculty"}]}}; {e} and 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 ${\mathbb T}^2$.

Item not freely available? Link broken?
Report a problem accessing this item