{
  "abstract": "We derive the secular Hamiltonian of a hierarchical triple from the 1PN ADM three-body Hamiltonian by a two-stage Lie-Hori-Deprit transformation carried out entirely in canonical variables, and use it to settle the relativistic cross terms left open in Paper I. A bookkeeping rule enumerates all contributions to each order in the 1PN parameter $δ$ and the hierarchy parameter $\\varepsilon$. The 1PN monopole interaction reduces to a renormalization of the inner mass by its binding energy, and the order $δ\\varepsilon^{3/2}$, at which Lim and Rodriguez (LR) found a libration term, is empty. The electric and magnetic 1PN $\\times$ quadrupole cross terms agree exactly with the effective-field-theory result of Kuntz, Serra, and Trincherini; for the magnetic term this is the first independent derivation. We give the next-to-leading order 1PN $\\times$ quadrupole term in closed form; it depends on the outer argument of pericenter $ω_2$. At quadrupole-squared order we obtain the complete secular Hamiltonian, which generates the secular equations of Will and of Conway and Will; its $2ω_2$ harmonic is a gauge term, removable even on the commensurability between the ZLK period and the nodal precession, by a generator given in closed form via Tremaine's fictitious time. The EIH and ADM descriptions coincide at 1PN order; the formulations differ only in the definition of the elements, for which we give the osculating-contact map. In secular integrations of LR's triples, the eccentricity enhancement with the octupole is controlled by the $ω_2$-dependent part of the next-to-leading order term and is two to three times that of Paper I, while LR's accelerated mergers remain absent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14505",
  "author_total": 1,
  "authors": [
    {
      "name": "Hideyoshi Arakida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f5m2fe1fq13px.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14505",
  "fulltext_url": null,
  "id": "f5m2fe1fq13px",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14505",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "openalex",
      "3103|Stellar, planetary, and galactic studies"
    ]
  ],
  "openalex": "W7221834547",
  "pdf_url": "https://arxiv.org/pdf/2610.14505",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f5m2fe1fq13px.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:31Z",
      "native_id": "2610.14505",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14505"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN $\\times$ quadrupole cross terms and of the quadrupole-squared term",
  "title_text": "Post-Newtonian secular dynamics of hierarchical triples II. Canonical derivation of the 1PN × quadrupole cross terms and of the quadrupole-squared term",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:31Z",
  "url": "https://research.draconitic.com/paper/f5m2fe1fq13px",
  "venue": "arXiv",
  "versions": []
}