{"paper":{"title":"Geometric Analysis of the Damped Harmonic Oscillator via the Lambert W Function","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Arpan Sharma, Bhargava Jogi, Ken Roberts, Muralikrishna Molli, S.R. Valluri","submitted_at":"2026-05-29T15:50:13Z","abstract_excerpt":"The underdamped harmonic oscillator is analyzed through the complex mapping $\\zeta = e^{-i\\varphi}we^{-w}$ with $w = \\beta t + i\\Omega t$, which transforms the dynamics into a logarithmic spiral. Within this framework, the displacement extrema correspond to crossings of the imaginary axis by $\\zeta(t)$, yielding the explicit times $t_n = (\\theta - \\varphi - \\pi/2 + n\\pi)/\\Omega$, where $\\theta = \\arctan(\\Omega/\\beta)$. The Lambert $W$ function provides closed-form solutions $t = -\\beta^{-1}W_k(-\\beta A/\\omega_0)$ for the times at which the spiral radius attains a given threshold $A$, covering "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.31456","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.31456/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"}