{"paper":{"title":"Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Andreas Fischle, Patrizio Neff","submitted_at":"2017-01-27T18:54:54Z","abstract_excerpt":"Let $F \\in {\\rm GL}^+(3)$ and consider the right polar decomposition $F = R_p(F)\\cdot U$ into an orthogonal factor $R_p(F) \\in {\\rm SO}(3)$ and a symmetric, positive definite factor $U(F) = \\sqrt{F^TF} \\in {\\rm Psym}(3)$. In 1940 Giuseppe Grioli proved that $$ {\\rm argmin}_{R \\in {\\rm SO}(3)} ||R^TF - 1{||}^2 \\quad=\\quad \\{\\,R_p(F)\\,\\} \\quad=\\quad {\\rm argmin}_{R \\in {\\rm SO}(3)} ||F - R{||}^2\\;. $$ This variational characterization of the orthogonal factor $R_p(F) \\in {\\rm SO}(n)$ holds in any dimension $n \\geq 2$ (a result due to Martins and Podio-Guidugli). In a similar spirit, we character"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08150","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}