{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:G3GYPFVIKIPFQ25ME7GFKEKTPH","short_pith_number":"pith:G3GYPFVI","canonical_record":{"source":{"id":"0707.0022","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2007-06-30T01:16:34Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"4fb0c37fd9effec4da5903321832f08be99e39e4aeee1ac4cfd77e5bfd308a45","abstract_canon_sha256":"8fced0afc989248c23ac5fb19fe367a69e98727d1b8dac9c4d2abedbca19a00e"},"schema_version":"1.0"},"canonical_sha256":"36cd8796a8521e586bac27cc55115379cdfd771dc9f9d3ba872b2c4f334cbbf2","source":{"kind":"arxiv","id":"0707.0022","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0707.0022","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"arxiv_version","alias_value":"0707.0022v1","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.0022","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_12","alias_value":"G3GYPFVIKIPF","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_16","alias_value":"G3GYPFVIKIPFQ25M","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_8","alias_value":"G3GYPFVI","created_at":"2026-06-03T22:06:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:G3GYPFVIKIPFQ25ME7GFKEKTPH","target":"record","payload":{"canonical_record":{"source":{"id":"0707.0022","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NA","submitted_at":"2007-06-30T01:16:34Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"4fb0c37fd9effec4da5903321832f08be99e39e4aeee1ac4cfd77e5bfd308a45","abstract_canon_sha256":"8fced0afc989248c23ac5fb19fe367a69e98727d1b8dac9c4d2abedbca19a00e"},"schema_version":"1.0"},"canonical_sha256":"36cd8796a8521e586bac27cc55115379cdfd771dc9f9d3ba872b2c4f334cbbf2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T22:06:01.408579Z","signature_b64":"dOsgizmn8fo6+w+GbYzRy5u8f1rnjAcHcajWIpCz++/TCLjtiWGEpZdxWi1jLFLkIMZ+c/x1I+FGTmy/IlE8Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36cd8796a8521e586bac27cc55115379cdfd771dc9f9d3ba872b2c4f334cbbf2","last_reissued_at":"2026-06-03T22:06:01.408093Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T22:06:01.408093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0707.0022","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-03T22:06:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ObL4DBEUpLkWQoc8snylltrbWusncEWYyIVb70UcrAfw5zp+VHpsSdAtNhbx9+081ZSpzkOJNMLzorwp1sNUCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:52:08.771183Z"},"content_sha256":"b48fc18bc181f6da4b466714607fd6799fa0c24e265ee0d993777b0302b0b58c","schema_version":"1.0","event_id":"sha256:b48fc18bc181f6da4b466714607fd6799fa0c24e265ee0d993777b0302b0b58c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:G3GYPFVIKIPFQ25ME7GFKEKTPH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lagrangian Mechanics and Variational Integrators on Two-Spheres","license":"","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Melvin Leok, N. Harris McClamroch, Taeyoung Lee","submitted_at":"2007-06-30T01:16:34Z","abstract_excerpt":"Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.0022","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0707.0022/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-03T22:06:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KxUpBd+6FWwWsCd/hicBhYT08MXWhGPG3BVfJNVE9qJddB1b4jg6I8DLcjbg8PceCOKGtYzb9hZ+GxzjL3ljBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:52:08.771955Z"},"content_sha256":"c623a8feff2e86546b9016ea63b569afc98607336761f50790ece12e342180ab","schema_version":"1.0","event_id":"sha256:c623a8feff2e86546b9016ea63b569afc98607336761f50790ece12e342180ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/bundle.json","state_url":"https://pith.science/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/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":"2026-06-10T23:52:08Z","links":{"resolver":"https://pith.science/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH","bundle":"https://pith.science/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/bundle.json","state":"https://pith.science/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G3GYPFVIKIPFQ25ME7GFKEKTPH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:G3GYPFVIKIPFQ25ME7GFKEKTPH","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":"8fced0afc989248c23ac5fb19fe367a69e98727d1b8dac9c4d2abedbca19a00e","cross_cats_sorted":["cs.NA"],"license":"","primary_cat":"math.NA","submitted_at":"2007-06-30T01:16:34Z","title_canon_sha256":"4fb0c37fd9effec4da5903321832f08be99e39e4aeee1ac4cfd77e5bfd308a45"},"schema_version":"1.0","source":{"id":"0707.0022","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0707.0022","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"arxiv_version","alias_value":"0707.0022v1","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0707.0022","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_12","alias_value":"G3GYPFVIKIPF","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_16","alias_value":"G3GYPFVIKIPFQ25M","created_at":"2026-06-03T22:06:01Z"},{"alias_kind":"pith_short_8","alias_value":"G3GYPFVI","created_at":"2026-06-03T22:06:01Z"}],"graph_snapshots":[{"event_id":"sha256:c623a8feff2e86546b9016ea63b569afc98607336761f50790ece12e342180ab","target":"graph","created_at":"2026-06-03T22:06:01Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/0707.0022/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since t","authors_text":"Melvin Leok, N. Harris McClamroch, Taeyoung Lee","cross_cats":["cs.NA"],"headline":"","license":"","primary_cat":"math.NA","submitted_at":"2007-06-30T01:16:34Z","title":"Lagrangian Mechanics and Variational Integrators on Two-Spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0707.0022","kind":"arxiv","version":1},"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:b48fc18bc181f6da4b466714607fd6799fa0c24e265ee0d993777b0302b0b58c","target":"record","created_at":"2026-06-03T22:06:01Z","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":"8fced0afc989248c23ac5fb19fe367a69e98727d1b8dac9c4d2abedbca19a00e","cross_cats_sorted":["cs.NA"],"license":"","primary_cat":"math.NA","submitted_at":"2007-06-30T01:16:34Z","title_canon_sha256":"4fb0c37fd9effec4da5903321832f08be99e39e4aeee1ac4cfd77e5bfd308a45"},"schema_version":"1.0","source":{"id":"0707.0022","kind":"arxiv","version":1}},"canonical_sha256":"36cd8796a8521e586bac27cc55115379cdfd771dc9f9d3ba872b2c4f334cbbf2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36cd8796a8521e586bac27cc55115379cdfd771dc9f9d3ba872b2c4f334cbbf2","first_computed_at":"2026-06-03T22:06:01.408093Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-03T22:06:01.408093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dOsgizmn8fo6+w+GbYzRy5u8f1rnjAcHcajWIpCz++/TCLjtiWGEpZdxWi1jLFLkIMZ+c/x1I+FGTmy/IlE8Dg==","signature_status":"signed_v1","signed_at":"2026-06-03T22:06:01.408579Z","signed_message":"canonical_sha256_bytes"},"source_id":"0707.0022","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b48fc18bc181f6da4b466714607fd6799fa0c24e265ee0d993777b0302b0b58c","sha256:c623a8feff2e86546b9016ea63b569afc98607336761f50790ece12e342180ab"],"state_sha256":"a89887538974151b9a8838f4cf3415e9e73639694a1cfbe6239deb0294c2fd32"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nx/pW+VX03vjTyGIiTZ+vBjPX5b2QTkKubqUXcKE9bgWYIm8JbCQVrM6ai8aNWp4buNYdC2UlRRNyq1vxI4OCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:52:08.776433Z","bundle_sha256":"9c7300c72d1454921a1b73e068ccfca1da8385521f63910412e4932ceee724e5"}}