{"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. 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