{
  "abstract": "The hyperbolic potential $$ V(r)=-\\fracα{(R^2-r^2)^2} $$ was identified by Olshanii as the negative-curvature counterpart of Newton's off-center circular-orbit problem. We first show that this form is not an isolated ansatz. For a planar radial natural Hamiltonian at fixed energy, if the Jacobi metric is regular at the force center and has nonzero constant Gaussian curvature, then, up to radial and overall scales, $$ E-V(r)=\\fracα{(R^2\\pm r^2)^2}. $$ Thus the spherical and hyperbolic off-center models are the two nonflat regular radial constant-curvature branches. We then complete the dynamics of the negative-curvature branch. At zero energy its Jacobi metric is a constant multiple of the Poincare disk metric. An explicit Runge--Lenz-type vector and the angular momentum form an on-shell $\\mathfrak{so}(2,1)$ moment map; their algebra gives the supporting-circle equation without integrating the equations of motion. The cotangent lift of circular inversion preserves the moment map and intertwines the exterior and punctured-interior flows after a positive time change. The singular circle is at infinite Jacobi distance but is reached in finite Newtonian time. Finally, requiring the magnetic two-form to preserve the full orientation-preserving hyperbolic isometry group uniquely forces it to be a constant multiple of hyperbolic area, hence $B(r)=-Q/(R^2-r^2)^2$ in disk coordinates. The resulting shifted $\\mathfrak{so}(2,1)$ Casimir gives the standard circle--horocycle--hypercycle transition at $Q^2=8mαR^2$, now as an explicit coupling statement in the off-center Newtonian system.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04521",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipesh Bhandari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e8bat9mkwtrzt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04521",
  "fulltext_url": null,
  "id": "e8bat9mkwtrzt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04521",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3103|Advanced Differential Geometry Research"
    ]
  ],
  "openalex": "W7167625355",
  "pdf_url": "https://arxiv.org/pdf/2607.04521",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e8bat9mkwtrzt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2607.04521",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04521"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, $SO(2,1)$ Symmetry, and Magnetic Uniqueness",
  "title_text": "Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, SO(2,1) Symmetry, and Magnetic Uniqueness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/e8bat9mkwtrzt",
  "venue": "arXiv",
  "versions": []
}