{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:VL7HPQLFETSNMCPWXR6V74KB6X","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":"d23f38fee96e99ff0d9a6ce647e4ebe0cf7b20ff3dc7dbeb49be5b4f287dfed3","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2009-03-17T06:55:12Z","title_canon_sha256":"f8f4f03336e298638d4398e548d1fb1c8aec913051ac2fc562f2055382026e42"},"schema_version":"1.0","source":{"id":"0903.2909","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.2909","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"arxiv_version","alias_value":"0903.2909v1","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.2909","created_at":"2026-05-18T02:14:27Z"},{"alias_kind":"pith_short_12","alias_value":"VL7HPQLFETSN","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"VL7HPQLFETSNMCPW","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"VL7HPQLF","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:7ab88dc19a8c1bbdc0999b10b62635c29e0094093a66607c7208e2b27ecdd348","target":"graph","created_at":"2026-05-18T02:14:27Z","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":"We first consider the Hamiltonian formulation of $n=3$ systems in general and show that all dynamical systems in ${\\mathbb R}^3$ are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We find the Poisson structures of a dynamical system recently given by Bender et al. Secondly, we show that all dynamical systems in ${\\mathbb R}^n$ are $(n-1)$-Hamiltonian. We give also an algorithm, similar to the case in ${\\mathbb R}^3$, to construct a rank two Poisson structure of dynamical systems in ${\\mathbb R}^n$. We give a classification of the dynamical ","authors_text":"Gusein Sh. Guseinov, Kostyantyn Zheltukhin, Metin Gurses","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2009-03-17T06:55:12Z","title":"Dynamical Systems and Poisson Structures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.2909","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":"sha256:89ac7a2a0e212ad7cbdd0bc6a7c8e5edc7e57aa62ccdc02bdc6951e62f8891f2","target":"record","created_at":"2026-05-18T02:14:27Z","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":"d23f38fee96e99ff0d9a6ce647e4ebe0cf7b20ff3dc7dbeb49be5b4f287dfed3","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"nlin.SI","submitted_at":"2009-03-17T06:55:12Z","title_canon_sha256":"f8f4f03336e298638d4398e548d1fb1c8aec913051ac2fc562f2055382026e42"},"schema_version":"1.0","source":{"id":"0903.2909","kind":"arxiv","version":1}},"canonical_sha256":"aafe77c16524e4d609f6bc7d5ff141f5fdc55c5a0422606b35b9c9a0cc24ea5d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aafe77c16524e4d609f6bc7d5ff141f5fdc55c5a0422606b35b9c9a0cc24ea5d","first_computed_at":"2026-05-18T02:14:27.846999Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:14:27.846999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OevgENklR8Yo3mO7Y9lKRvGtXiT6IzF7arKaCWkVS+PBwP3QCsZQ+FhgCTrnkUAh1AkkvOAuWipdsUbhmUoDDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:14:27.847490Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.2909","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89ac7a2a0e212ad7cbdd0bc6a7c8e5edc7e57aa62ccdc02bdc6951e62f8891f2","sha256:7ab88dc19a8c1bbdc0999b10b62635c29e0094093a66607c7208e2b27ecdd348"],"state_sha256":"fe8a0da1fe07b0555ca46a1618e3160ccc9f6d61f264d139af1e472419d1820e"}