{"paper":{"title":"A simple way of making a Hamiltonian system into a bi-Hamiltonian one","license":"","headline":"","cross_cats":["hep-th","math-ph","math.DG","math.MP","math.SG"],"primary_cat":"nlin.SI","authors_text":"A. Sergyeyev","submitted_at":"2003-10-13T15:43:56Z","abstract_excerpt":"Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_{\\tau}(P)$ along a vector field $\\tau$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_{\\tau}^2(P),P]=0$, where $[\\cdot,\\cdot]$ is the Schouten bracket. We further prove that if $\\dim\\ker P\\leq 1$ and $P$ is of locally constant rank, then all Poisson structures compatible with a given Poisson structure $P$ on a finite-dimensional manifold $M$ are locally of the form $L_{\\tau}(P)$, where $\\tau$ is a local vector field such that $L_{\\tau}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"nlin/0310012","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/nlin/0310012/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}