{"paper":{"title":"Lagrangian mechanics on centered semi-direct product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Henry O. Jacobs, Leonardo J. Colombo","submitted_at":"2013-03-15T19:55:20Z","abstract_excerpt":"There exists two types of semi-direct products between a Lie group $G$ and a vector space $V$.\n  The left semi-direct product, $G \\ltimes V$, can be constructed when $G$ is equipped with a left action on $V$.\n  Similarly, the right semi-direct product, $G \\rtimes V$, can be constructed when $G$ is equipped with a right action on $V$.\n  In this paper, we will construct a new type of semi-direct product, $G \\Join V$, which can be seen as the `sum' of a right and left semi-direct product.\n  We then parallel existing semi-direct product Euler-Poincar\\'{e} theory.\n  We find that the group multiplic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3883","kind":"arxiv","version":2},"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"}