{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:SH4WCGCPWNRWSPGMM2ZDW2I4D6","short_pith_number":"pith:SH4WCGCP","canonical_record":{"source":{"id":"0712.2766","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2007-12-17T16:38:16Z","cross_cats_sorted":["math.DG","math.MP"],"title_canon_sha256":"ae38865e8d759969ffcddbe81b1fdb4de167782a06e137a45bc7312699bd9211","abstract_canon_sha256":"2309199773436e6558cda85dd9ba89316cf6a1ac6c95eab721dcdb659ca6ffe3"},"schema_version":"1.0"},"canonical_sha256":"91f961184fb363693ccc66b23b691c1fad43baaf460a0fb61193ac82d09b339d","source":{"kind":"arxiv","id":"0712.2766","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.2766","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"arxiv_version","alias_value":"0712.2766v2","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2766","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"pith_short_12","alias_value":"SH4WCGCPWNRW","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"SH4WCGCPWNRWSPGM","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"SH4WCGCP","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:SH4WCGCPWNRWSPGMM2ZDW2I4D6","target":"record","payload":{"canonical_record":{"source":{"id":"0712.2766","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2007-12-17T16:38:16Z","cross_cats_sorted":["math.DG","math.MP"],"title_canon_sha256":"ae38865e8d759969ffcddbe81b1fdb4de167782a06e137a45bc7312699bd9211","abstract_canon_sha256":"2309199773436e6558cda85dd9ba89316cf6a1ac6c95eab721dcdb659ca6ffe3"},"schema_version":"1.0"},"canonical_sha256":"91f961184fb363693ccc66b23b691c1fad43baaf460a0fb61193ac82d09b339d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:04.304378Z","signature_b64":"xCJVls20uUt+6u/y0e0Was2q6w1m9t9QrpTzhqdUSHKmZ5eoktSh/FST0qfgJYUDni1VlfU6Di7Bgc8C3f6GAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91f961184fb363693ccc66b23b691c1fad43baaf460a0fb61193ac82d09b339d","last_reissued_at":"2026-05-18T04:08:04.303860Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:04.303860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0712.2766","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gc/KphFd1jQSwixK8/IHufZHe34KYCD7BuhtPKSEKATrVJ71io8oq3VsontCfr5c9MedQGLT1oHI0+TV4EzXBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:39:56.311098Z"},"content_sha256":"a1ad8c55a838198137c231b8592a9c81c68534ed0f525bb8db6b5ccb175a7e80","schema_version":"1.0","event_id":"sha256:a1ad8c55a838198137c231b8592a9c81c68534ed0f525bb8db6b5ccb175a7e80"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:SH4WCGCPWNRWSPGMM2ZDW2I4D6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Variational calculus with constraints on general algebroids","license":"","headline":"","cross_cats":["math.DG","math.MP"],"primary_cat":"math-ph","authors_text":"Janusz Grabowski, Katarzyna Grabowska","submitted_at":"2007-12-17T16:38:16Z","abstract_excerpt":"Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2766","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"huGiYuYRlySZyj2TXQ/xIk41vkFgYKqp9k+1P4PmgJynucsELPS+x9dEJh9uf7zUpsMfeWF0llAPzIjy/bbNAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T09:39:56.311465Z"},"content_sha256":"b46a58c2f4689fb38a0e7bb26b4becfcfe1508e75044698112e86f0a4fb562c0","schema_version":"1.0","event_id":"sha256:b46a58c2f4689fb38a0e7bb26b4becfcfe1508e75044698112e86f0a4fb562c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/bundle.json","state_url":"https://pith.science/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T09:39:56Z","links":{"resolver":"https://pith.science/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6","bundle":"https://pith.science/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/bundle.json","state":"https://pith.science/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SH4WCGCPWNRWSPGMM2ZDW2I4D6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:SH4WCGCPWNRWSPGMM2ZDW2I4D6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2309199773436e6558cda85dd9ba89316cf6a1ac6c95eab721dcdb659ca6ffe3","cross_cats_sorted":["math.DG","math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2007-12-17T16:38:16Z","title_canon_sha256":"ae38865e8d759969ffcddbe81b1fdb4de167782a06e137a45bc7312699bd9211"},"schema_version":"1.0","source":{"id":"0712.2766","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.2766","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"arxiv_version","alias_value":"0712.2766v2","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2766","created_at":"2026-05-18T04:08:04Z"},{"alias_kind":"pith_short_12","alias_value":"SH4WCGCPWNRW","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"SH4WCGCPWNRWSPGM","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"SH4WCGCP","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:b46a58c2f4689fb38a0e7bb26b4becfcfe1508e75044698112e86f0a4fb562c0","target":"graph","created_at":"2026-05-18T04:08:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.","authors_text":"Janusz Grabowski, Katarzyna Grabowska","cross_cats":["math.DG","math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2007-12-17T16:38:16Z","title":"Variational calculus with constraints on general algebroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2766","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a1ad8c55a838198137c231b8592a9c81c68534ed0f525bb8db6b5ccb175a7e80","target":"record","created_at":"2026-05-18T04:08:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2309199773436e6558cda85dd9ba89316cf6a1ac6c95eab721dcdb659ca6ffe3","cross_cats_sorted":["math.DG","math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2007-12-17T16:38:16Z","title_canon_sha256":"ae38865e8d759969ffcddbe81b1fdb4de167782a06e137a45bc7312699bd9211"},"schema_version":"1.0","source":{"id":"0712.2766","kind":"arxiv","version":2}},"canonical_sha256":"91f961184fb363693ccc66b23b691c1fad43baaf460a0fb61193ac82d09b339d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"91f961184fb363693ccc66b23b691c1fad43baaf460a0fb61193ac82d09b339d","first_computed_at":"2026-05-18T04:08:04.303860Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:04.303860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xCJVls20uUt+6u/y0e0Was2q6w1m9t9QrpTzhqdUSHKmZ5eoktSh/FST0qfgJYUDni1VlfU6Di7Bgc8C3f6GAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:04.304378Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.2766","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1ad8c55a838198137c231b8592a9c81c68534ed0f525bb8db6b5ccb175a7e80","sha256:b46a58c2f4689fb38a0e7bb26b4becfcfe1508e75044698112e86f0a4fb562c0"],"state_sha256":"c2b19cbbd930d5da5cac666642f7b485b1e46bfc3f8b9dd063d45d51b3b47c91"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O531PXOE8Z1fMBlpYMShUQOgp8qMecReDwfkxwKBDopk6B5xFyOG0t+393CZ0vJmTc2FdaXvaXnUYepqf6WYBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T09:39:56.313412Z","bundle_sha256":"00e788c37397d3334e32833abf1ad1e56d0764c6bf4fc566b64f3d9f28f8c63e"}}