{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:6IQO4YTYQ7HNWNL5PA3WCGNTUX","short_pith_number":"pith:6IQO4YTY","schema_version":"1.0","canonical_sha256":"f220ee627887cedb357d78376119b3a5c550438830ac467ba7a9d82bafd128da","source":{"kind":"arxiv","id":"2605.15679","version":1},"attestation_state":"computed","paper":{"title":"Spectral separation of variables from equivalent Lagrangian systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Requiring two quadratic Lagrangians to produce the same equations of motion imposes a commutation condition that spectrally decomposes the configuration space and decouples the dynamics.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Mattia Scomparin","submitted_at":"2026-05-15T07:01:35Z","abstract_excerpt":"We investigate the dynamical equivalence of quadratic Lagrangians and its relation to separation of variables. We show that requiring two quadratic Lagrangians to generate the same Euler--Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space. The equations of motion then decouple into independent subsystems: generically in block-separated form, and comp"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.15679","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-05-15T07:01:35Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"8edc86f27dcab524da00d6d9e40e454f493ce909bb6c46c4943c99cb45dc540f","abstract_canon_sha256":"81631051bb8e8cb3ace28d925648991259f3e72c4e78f07f4baa195d87c6afe7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:01:11.985441Z","signature_b64":"anL5AfmhYLHgUSHRCOvW4idADlHJ60rGzFBYkbJkcfWzPmjb5IdXdTQacx4r48C1k157acMJQuZ3Huac5YO8Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f220ee627887cedb357d78376119b3a5c550438830ac467ba7a9d82bafd128da","last_reissued_at":"2026-05-20T00:01:11.984610Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:01:11.984610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral separation of variables from equivalent Lagrangian systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Requiring two quadratic Lagrangians to produce the same equations of motion imposes a commutation condition that spectrally decomposes the configuration space and decouples the dynamics.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Mattia Scomparin","submitted_at":"2026-05-15T07:01:35Z","abstract_excerpt":"We investigate the dynamical equivalence of quadratic Lagrangians and its relation to separation of variables. We show that requiring two quadratic Lagrangians to generate the same Euler--Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space. The equations of motion then decouple into independent subsystems: generically in block-separated form, and comp"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Requiring two quadratic Lagrangians to generate the same Euler-Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The kinetic matrices are constant and symmetric, which is required for the compatibility condition to reduce to a commutation relation with the Hessian (as stated in the abstract for the constant symmetric case).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Dynamical equivalence of quadratic Lagrangians implies a commutation relation with the potential Hessian that yields orthogonal spectral decomposition and decoupled equations of motion.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Requiring two quadratic Lagrangians to produce the same equations of motion imposes a commutation condition that spectrally decomposes the configuration space and decouples the dynamics.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ada457774de0063d606ee1ac329fc519c769a8f5c1ffef6cc6deb388211d5971"},"source":{"id":"2605.15679","kind":"arxiv","version":1},"verdict":{"id":"7abf0d83-831e-4ca1-8a6d-612be4cf2620","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:40:42.607134Z","strongest_claim":"Requiring two quadratic Lagrangians to generate the same Euler-Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space.","one_line_summary":"Dynamical equivalence of quadratic Lagrangians implies a commutation relation with the potential Hessian that yields orthogonal spectral decomposition and decoupled equations of motion.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The kinetic matrices are constant and symmetric, which is required for the compatibility condition to reduce to a commutation relation with the Hessian (as stated in the abstract for the constant symmetric case).","pith_extraction_headline":"Requiring two quadratic Lagrangians to produce the same equations of motion imposes a commutation condition that spectrally decomposes the configuration space and decouples the dynamics."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15679/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:01:19.238210Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:51:29.314582Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:33:29.858989Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:21:56.054990Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"43440d414481a1dc1b1ee2d9bbc555db546dac44f71147d9348bf3a88ccf82db"},"references":{"count":19,"sample":[{"doi":"","year":1997,"title":"Sergio Benenti , Intrinsic characterization of the variable separation in t he Hamilton–Jacobi equation , Journal of Mathematical Physics 38 (1997), no. 12, 6578–6602","work_id":"2185feb8-48a7-4a4d-b07b-39a28fe5b6d2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"Sergio Benenti, Claudia Chanu, and Giovanni Rastelli , Variable sepa- ration for natural hamiltonians with scalar and vector pote ntials on riemannian manifolds, Journal of Mathematical Physics 42 (20","work_id":"34554f29-8a43-4b5f-9691-5252ba88930d","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Sergio Benenti, Claudia Chanu, and Giovanni Rastelli , Variable- separation theory for the null Hamilton–Jacobi equation , Journal of Mathemati- cal Physics 46 (2005), 042901","work_id":"b42f9d01-bd73-4ffb-9c87-0c946af6dc37","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"B/suppress laszak and S","work_id":"89689e6e-e6c7-4927-aa3f-6f3dc78759eb","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1993,"title":"C. M. Cosgrove and G. Scoufis , Painlev´ e classiﬁcation of a class of dif- ferential equations of the second order and second degree , Studies in Applied Mathematics 88 (1993), no. 1, 25–87","work_id":"36e89b4a-1cc6-4315-9758-4ea853b73136","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"283c8dda4d8bfc7cc9db030a86f98b1f1db941d8f23fe2734abc9ecdd54f5866","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"905f92643559764e5bc922e34db0364fe6d3fb772b880ab0921c82367e783e21"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.15679","created_at":"2026-05-20T00:01:11.984720+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.15679v1","created_at":"2026-05-20T00:01:11.984720+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.15679","created_at":"2026-05-20T00:01:11.984720+00:00"},{"alias_kind":"pith_short_12","alias_value":"6IQO4YTYQ7HN","created_at":"2026-05-20T00:01:11.984720+00:00"},{"alias_kind":"pith_short_16","alias_value":"6IQO4YTYQ7HNWNL5","created_at":"2026-05-20T00:01:11.984720+00:00"},{"alias_kind":"pith_short_8","alias_value":"6IQO4YTY","created_at":"2026-05-20T00:01:11.984720+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX","json":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX.json","graph_json":"https://pith.science/api/pith-number/6IQO4YTYQ7HNWNL5PA3WCGNTUX/graph.json","events_json":"https://pith.science/api/pith-number/6IQO4YTYQ7HNWNL5PA3WCGNTUX/events.json","paper":"https://pith.science/paper/6IQO4YTY"},"agent_actions":{"view_html":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX","download_json":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX.json","view_paper":"https://pith.science/paper/6IQO4YTY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.15679&json=true","fetch_graph":"https://pith.science/api/pith-number/6IQO4YTYQ7HNWNL5PA3WCGNTUX/graph.json","fetch_events":"https://pith.science/api/pith-number/6IQO4YTYQ7HNWNL5PA3WCGNTUX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX/action/storage_attestation","attest_author":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX/action/author_attestation","sign_citation":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX/action/citation_signature","submit_replication":"https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX/action/replication_record"}},"created_at":"2026-05-20T00:01:11.984720+00:00","updated_at":"2026-05-20T00:01:11.984720+00:00"}