{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:38:58Z","timestamp":1760233138575,"version":"build-2065373602"},"reference-count":44,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T00:00:00Z","timestamp":1671667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"T\u00dcB\u0130TAK (the Scientific and Technological Research Council of Turkey)","award":["117F426"],"award-info":[{"award-number":["117F426"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler\u2013Poincar\u00e9 flows on such spaces are decomposed into two subdynamics: one is the Euler\u2013Poincar\u00e9 formulation of isentropic fluid flows, and the other one corresponds with Euler\u2013Poincar\u00e9 equations on contravariant tensors of order n\u2a7e2.<\/jats:p>","DOI":"10.3390\/sym15010023","type":"journal-article","created":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T02:31:11Z","timestamp":1671676271000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Decomposing Euler\u2013Poincar\u00e9 Flow on the Space of Hamiltonian Vector Fields"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6766-0287","authenticated-orcid":false,"given":"O\u011ful","family":"Esen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Gebze Technical University, Gebze 41400, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8643-144X","authenticated-orcid":false,"given":"Javier","family":"De Lucas","sequence":"additional","affiliation":[{"name":"Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cristina Sardon","family":"Mu\u00f1oz","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Universidad Polit\u00e9cnica de Madrid, C\/Jos\u00e9 Guti\u00e9rrez Abascal, 2, 28006 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcin","family":"Zaj\u0105c","sequence":"additional","affiliation":[{"name":"Department of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,22]]},"reference":[{"key":"ref_1","unstructured":"de Le\u00f3n, M., and Rodrigues, P.R. (2011). Methods of Differential Geometry in Analytical Mechanics, Elsevier."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Holm, D.D. (2008). 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