{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:HRHCGEK2U2GJGO3AZHBJOAB26I","short_pith_number":"pith:HRHCGEK2","canonical_record":{"source":{"id":"1309.0149","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-31T20:01:38Z","cross_cats_sorted":[],"title_canon_sha256":"7565ed701e570aa907919b6d8b710e881671e30d8b744f4bd711e00ddcbdc883","abstract_canon_sha256":"95d45a63b7af90c61fb17a67e7a16f97588b128c32eea66c8b3513f2766fc58c"},"schema_version":"1.0"},"canonical_sha256":"3c4e23115aa68c933b60c9c297003af238d4d1dd8dd50892ba0f72f705043dad","source":{"kind":"arxiv","id":"1309.0149","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0149","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0149v1","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0149","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"pith_short_12","alias_value":"HRHCGEK2U2GJ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HRHCGEK2U2GJGO3A","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HRHCGEK2","created_at":"2026-05-18T12:27:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:HRHCGEK2U2GJGO3AZHBJOAB26I","target":"record","payload":{"canonical_record":{"source":{"id":"1309.0149","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-31T20:01:38Z","cross_cats_sorted":[],"title_canon_sha256":"7565ed701e570aa907919b6d8b710e881671e30d8b744f4bd711e00ddcbdc883","abstract_canon_sha256":"95d45a63b7af90c61fb17a67e7a16f97588b128c32eea66c8b3513f2766fc58c"},"schema_version":"1.0"},"canonical_sha256":"3c4e23115aa68c933b60c9c297003af238d4d1dd8dd50892ba0f72f705043dad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:22.290688Z","signature_b64":"BVTdCVssRIcdobpoQpaE6WSsHtKjH/1dtczS9yrMCYKFHEHcrJQVoZrnJ+vd+bUr4uV0N8nhTrpn5FW4gp+0AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c4e23115aa68c933b60c9c297003af238d4d1dd8dd50892ba0f72f705043dad","last_reissued_at":"2026-05-18T02:18:22.289926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:22.289926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.0149","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":"2026-05-18T02:18:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YYbAb2VknsN+eVsF3+r5OSFrRjbxBVOpdfAGHmRlE4F4QGNfsTK+zOcNLXNaO+qvHavyRDE3+uHu2spqiFFHCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:09:10.741456Z"},"content_sha256":"1b2f425df9cee0a70a8808577a6b58f66c286eabfab322922288628aef2fa594","schema_version":"1.0","event_id":"sha256:1b2f425df9cee0a70a8808577a6b58f66c286eabfab322922288628aef2fa594"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:HRHCGEK2U2GJGO3AZHBJOAB26I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lectures on the free period Lagrangian action functional","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Alberto Abbondandolo","submitted_at":"2013-08-31T20:01:38Z","abstract_excerpt":"In this expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. We use a variational approach, by studying how the behavior of the free period Lagrangian action functional changes when the energy crosses certain values, known as the Ma\\~n\\'e critical values."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0149","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":""},"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":"2026-05-18T02:18:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TtSPPsyffBjviHKKu1e0P7rfDHCIvmpdIX3NKTqcQqeyGoCkN1cjg1AlgantA9eTr9mTwU7l6CqW2FaGx156AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T12:09:10.741802Z"},"content_sha256":"d465da6393edcdebb05b4458edb56f7a09967f0793e7afc428fa1c8cebe2e7bf","schema_version":"1.0","event_id":"sha256:d465da6393edcdebb05b4458edb56f7a09967f0793e7afc428fa1c8cebe2e7bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/bundle.json","state_url":"https://pith.science/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/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":"2026-06-02T12:09:10Z","links":{"resolver":"https://pith.science/pith/HRHCGEK2U2GJGO3AZHBJOAB26I","bundle":"https://pith.science/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/bundle.json","state":"https://pith.science/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HRHCGEK2U2GJGO3AZHBJOAB26I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:HRHCGEK2U2GJGO3AZHBJOAB26I","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":"95d45a63b7af90c61fb17a67e7a16f97588b128c32eea66c8b3513f2766fc58c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-31T20:01:38Z","title_canon_sha256":"7565ed701e570aa907919b6d8b710e881671e30d8b744f4bd711e00ddcbdc883"},"schema_version":"1.0","source":{"id":"1309.0149","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.0149","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"arxiv_version","alias_value":"1309.0149v1","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0149","created_at":"2026-05-18T02:18:22Z"},{"alias_kind":"pith_short_12","alias_value":"HRHCGEK2U2GJ","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_16","alias_value":"HRHCGEK2U2GJGO3A","created_at":"2026-05-18T12:27:46Z"},{"alias_kind":"pith_short_8","alias_value":"HRHCGEK2","created_at":"2026-05-18T12:27:46Z"}],"graph_snapshots":[{"event_id":"sha256:d465da6393edcdebb05b4458edb56f7a09967f0793e7afc428fa1c8cebe2e7bf","target":"graph","created_at":"2026-05-18T02:18:22Z","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 expository article we study the question of the existence of periodic orbits of prescribed energy for classical Hamiltonian systems on compact configuration spaces. We use a variational approach, by studying how the behavior of the free period Lagrangian action functional changes when the energy crosses certain values, known as the Ma\\~n\\'e critical values.","authors_text":"Alberto Abbondandolo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-31T20:01:38Z","title":"Lectures on the free period Lagrangian action functional"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0149","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":"sha256:1b2f425df9cee0a70a8808577a6b58f66c286eabfab322922288628aef2fa594","target":"record","created_at":"2026-05-18T02:18:22Z","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":"95d45a63b7af90c61fb17a67e7a16f97588b128c32eea66c8b3513f2766fc58c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-08-31T20:01:38Z","title_canon_sha256":"7565ed701e570aa907919b6d8b710e881671e30d8b744f4bd711e00ddcbdc883"},"schema_version":"1.0","source":{"id":"1309.0149","kind":"arxiv","version":1}},"canonical_sha256":"3c4e23115aa68c933b60c9c297003af238d4d1dd8dd50892ba0f72f705043dad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c4e23115aa68c933b60c9c297003af238d4d1dd8dd50892ba0f72f705043dad","first_computed_at":"2026-05-18T02:18:22.289926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:22.289926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BVTdCVssRIcdobpoQpaE6WSsHtKjH/1dtczS9yrMCYKFHEHcrJQVoZrnJ+vd+bUr4uV0N8nhTrpn5FW4gp+0AA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:22.290688Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.0149","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b2f425df9cee0a70a8808577a6b58f66c286eabfab322922288628aef2fa594","sha256:d465da6393edcdebb05b4458edb56f7a09967f0793e7afc428fa1c8cebe2e7bf"],"state_sha256":"f80c3c002fe0155a91b028b11d9e9b9c2944e3142cf5b6f53bf96bbb2d489e65"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6ZtSnBgUPyjSQVJBSp49//sblvNA710qKu88FD8/QfdBL9lulPtEaAgejbCm6EyHfNwL+I1WccAZ/j2VfGVVDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T12:09:10.743960Z","bundle_sha256":"0e1e00c111fd00837b8ae1f52a860011954fe1454a1dcd6a7f7d8f8ddc14cf06"}}