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The canonical $\\left(\\rho ,\\eta \\right) $\\emph{-}semi(spray) associated to a mechanical $\\left(\\rho ,\\eta \\right) $-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical $(\\rho ,\\eta)$% -systems are the spaces necessary to develop a new Lagrangian formalism. We obtain the $(\\rho "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.2131","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-10-06T05:53:44Z","cross_cats_sorted":["math.DS","math.MP"],"title_canon_sha256":"a30d939fade24b56ef8387e8cc23c9b42c72eaf278ef2cf8c4c801704e201137","abstract_canon_sha256":"456df73f82e0cf38e4d71d938c55ab1d5ef6013ade28f84db444af24d7625589"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:59.242510Z","signature_b64":"DRVougVrvitvNljBTWRlR+s/0Ru+u/Zc84WjAnaKcGAosmGxfJ4YyJAf1ygm7xYEIfvuJZ37ihiWo1Efr2VnAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6dbdead690197339daf12995fea32292061d2a70b7fa1ef9939f1d2d64a5013","last_reissued_at":"2026-05-18T03:10:59.241752Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:59.241752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mechanical Systems in the Generalized Lie Algebroids Framework","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.MP"],"primary_cat":"math-ph","authors_text":"Constantin M. Arcus","submitted_at":"2013-10-06T05:53:44Z","abstract_excerpt":"\\emph{Mechanical systems} called by use, \\emph{mechanical}$\\left(\\rho ,\\eta\\right) $\\emph{-systems, Lagrange mechanical}$\\left(\\rho ,\\eta \\right) $\\emph{-systems} or \\emph{Finsler mechanical}$\\left(\\rho ,\\eta \\right) $\\emph{-systems} are presented. The canonical $\\left(\\rho ,\\eta \\right) $\\emph{-}semi(spray) associated to a mechanical $\\left(\\rho ,\\eta \\right) $-system is obtained. New and important results are obtained in the particular case of Lie algebroids. The Lagrange mechanical $(\\rho ,\\eta)$% -systems are the spaces necessary to develop a new Lagrangian formalism. 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