{"paper":{"title":"Completeness of the cubic and quartic H\\'enon-Heiles Hamiltonians","license":"","headline":"","cross_cats":[],"primary_cat":"nlin.SI","authors_text":"Caroline Verhoeven, Micheline Musette, Robert Conte","submitted_at":"2005-07-07T13:49:05Z","abstract_excerpt":"The quartic H\\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\\Omega_1 Q_1^2+\\Omega_2 Q_2^2)/2\n +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4\n +(1/2)(\\alpha/Q_1^2+\\beta/Q_2^2) - \\gamma Q_1$ passes the Painlev\\'e test for only four sets of values of the constants. Only one of these, identical to the traveling wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the three others are not yet integrated in the generic case $(\\alpha,\\beta,\\gamma)\\not=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"nlin/0507011","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"}