{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:22W6MA3KGF3OR5ZUR3EZHU3QZL","short_pith_number":"pith:22W6MA3K","canonical_record":{"source":{"id":"1412.2719","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-08T20:20:34Z","cross_cats_sorted":["math.DG","math.MP"],"title_canon_sha256":"7bb4722c2f481e7b88c92ee13aaf5fa95436f7d650cb8108f7b1e493b7fdcfe0","abstract_canon_sha256":"c7f98517f32b428f608c6be91d70d882713216518a3da43b845b20c70a93a5e2"},"schema_version":"1.0"},"canonical_sha256":"d6ade6036a3176e8f7348ec993d370caeeb1d3220274a1994fc4415ff2732a85","source":{"kind":"arxiv","id":"1412.2719","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2719","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2719v2","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2719","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"pith_short_12","alias_value":"22W6MA3KGF3O","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"22W6MA3KGF3OR5ZU","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"22W6MA3K","created_at":"2026-05-18T12:28:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:22W6MA3KGF3OR5ZUR3EZHU3QZL","target":"record","payload":{"canonical_record":{"source":{"id":"1412.2719","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-08T20:20:34Z","cross_cats_sorted":["math.DG","math.MP"],"title_canon_sha256":"7bb4722c2f481e7b88c92ee13aaf5fa95436f7d650cb8108f7b1e493b7fdcfe0","abstract_canon_sha256":"c7f98517f32b428f608c6be91d70d882713216518a3da43b845b20c70a93a5e2"},"schema_version":"1.0"},"canonical_sha256":"d6ade6036a3176e8f7348ec993d370caeeb1d3220274a1994fc4415ff2732a85","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:10.998500Z","signature_b64":"YyA6tMAmpu9ynWCT+F6nkM1y0BBspyfFA6VY60C+z61L2HK2VPz/OmhJQShmuN2eU9iVCauiz6a0Jb+wlnJBBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6ade6036a3176e8f7348ec993d370caeeb1d3220274a1994fc4415ff2732a85","last_reissued_at":"2026-05-18T01:24:10.997802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:10.997802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.2719","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":"2026-05-18T01:24:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2H0JzgQB8+v92UQzf9Fuf5PPSoylA3WROv18M1Wdiy/Dp4RvbmMIA9eELZ2c21PS3D+m8/g75UcDda/FmK62Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:32:33.240064Z"},"content_sha256":"04634f3554c2e035adfa189183eee8268de217e29afce65f77a37d843faacfa1","schema_version":"1.0","event_id":"sha256:04634f3554c2e035adfa189183eee8268de217e29afce65f77a37d843faacfa1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:22W6MA3KGF3OR5ZUR3EZHU3QZL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Higher order mechanics on graded bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MP"],"primary_cat":"math-ph","authors_text":"Andrew James Bruce, Janusz Grabowski, Katarzyna Grabowska","submitted_at":"2014-12-08T20:20:34Z","abstract_excerpt":"In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie grou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2719","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":"2026-05-18T01:24:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DAo6aEgzwNjYY2t98eSwJaCJ77qhtp+hQ8KJ/D4EPMLTz8nDOqrBleUXftwag0YXkR1HFzrSyfwv3iFHBS2sDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T08:32:33.240430Z"},"content_sha256":"df71385882724ed324be90b5763b01c232fe32c7387eb23cd7cc836f363292c9","schema_version":"1.0","event_id":"sha256:df71385882724ed324be90b5763b01c232fe32c7387eb23cd7cc836f363292c9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/bundle.json","state_url":"https://pith.science/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/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":"2026-06-25T08:32:33Z","links":{"resolver":"https://pith.science/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL","bundle":"https://pith.science/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/bundle.json","state":"https://pith.science/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/22W6MA3KGF3OR5ZUR3EZHU3QZL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:22W6MA3KGF3OR5ZUR3EZHU3QZL","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":"c7f98517f32b428f608c6be91d70d882713216518a3da43b845b20c70a93a5e2","cross_cats_sorted":["math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-08T20:20:34Z","title_canon_sha256":"7bb4722c2f481e7b88c92ee13aaf5fa95436f7d650cb8108f7b1e493b7fdcfe0"},"schema_version":"1.0","source":{"id":"1412.2719","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2719","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2719v2","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2719","created_at":"2026-05-18T01:24:10Z"},{"alias_kind":"pith_short_12","alias_value":"22W6MA3KGF3O","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"22W6MA3KGF3OR5ZU","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"22W6MA3K","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:df71385882724ed324be90b5763b01c232fe32c7387eb23cd7cc836f363292c9","target":"graph","created_at":"2026-05-18T01:24:10Z","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":"In this paper we develop a geometric approach to higher order mechanics on graded bundles in both, the Lagrangian and Hamiltonian formalism, via the recently discovered weighted algebroids. We present the corresponding Tulczyjew triple for this higher order situation and derive in this framework the phase equations from an arbitrary (also singular) Lagrangian or Hamiltonian, as well as the Euler-Lagrange equations. As important examples, we geometrically derive the classical higher order Euler-Lagrange equations and analogous reduced equations for invariant higher order Lagrangians on Lie grou","authors_text":"Andrew James Bruce, Janusz Grabowski, Katarzyna Grabowska","cross_cats":["math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-08T20:20:34Z","title":"Higher order mechanics on graded bundles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2719","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":"sha256:04634f3554c2e035adfa189183eee8268de217e29afce65f77a37d843faacfa1","target":"record","created_at":"2026-05-18T01:24:10Z","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":"c7f98517f32b428f608c6be91d70d882713216518a3da43b845b20c70a93a5e2","cross_cats_sorted":["math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-12-08T20:20:34Z","title_canon_sha256":"7bb4722c2f481e7b88c92ee13aaf5fa95436f7d650cb8108f7b1e493b7fdcfe0"},"schema_version":"1.0","source":{"id":"1412.2719","kind":"arxiv","version":2}},"canonical_sha256":"d6ade6036a3176e8f7348ec993d370caeeb1d3220274a1994fc4415ff2732a85","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6ade6036a3176e8f7348ec993d370caeeb1d3220274a1994fc4415ff2732a85","first_computed_at":"2026-05-18T01:24:10.997802Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:10.997802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YyA6tMAmpu9ynWCT+F6nkM1y0BBspyfFA6VY60C+z61L2HK2VPz/OmhJQShmuN2eU9iVCauiz6a0Jb+wlnJBBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:10.998500Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.2719","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:04634f3554c2e035adfa189183eee8268de217e29afce65f77a37d843faacfa1","sha256:df71385882724ed324be90b5763b01c232fe32c7387eb23cd7cc836f363292c9"],"state_sha256":"3298e7043efdcc40afc97fb0bc487bcda2c731db4a5c4108f79f27e1086c6477"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MhV2MZUZxv0E0BRKY4FsgdMxTLvkOmEs7vCGr/qTnIWyWjbHCEjufC92Gg8Jguwxc7i1soGxHph+XVznB82WAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T08:32:33.242376Z","bundle_sha256":"d0734ff7c1d8103bddd3df5e91f9aa69f7f2d41af5726f1173bee5611aabfc76"}}