{"paper":{"title":"Existence of isotropic complete solutions of the $\\Pi$-Hamilton-Jacobi equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Sergio Grillo","submitted_at":"2019-02-06T17:16:08Z","abstract_excerpt":"Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and a fibration $\\Pi:M\\rightarrow N$. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the $\\Pi$-HJE for $X$, whose unknown is a section $\\sigma:N\\rightarrow M$ of $\\Pi$. The standard HJE is obtained when the phase space $M$ is a cotangent bundle $T^{*}Q$ (with its canonical symplectic form), $\\Pi$ is the canonical projection $\\pi_{Q}:T^{*}Q\\rightarrow Q$ and the unknown is a closed $1$-form $\\mathsf{d}W:Q\\rightarrow T^{*}Q$. The function $W$ is called Hamilton's characteri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.02280","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"}