Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, SO(2,1) Symmetry, and Magnetic Uniqueness
Abstract
The hyperbolic potential V(r)=−α/((R2−r2)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)=α/((R2± r2)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 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/(R2−r2)2 in disk coordinates. The resulting shifted so(2,1) Casimir gives the standard circle--horocycle--hypercycle transition at Q2=8mαR2, now as an explicit coupling statement in the off-center Newtonian system.
Related
- Stability and uniqueness of minimal disks in non-constant curvature
- Rigidity of conformal metrics with constant sixth-order Q-curvature
- Constant mean curvature disks with circular boundary in the $3$-sphere and in hyperbolic space
- Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set
- Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity
BibTeX
@misc{bhandari2026hyperbolic,
title = {{Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, $SO(2,1)$ Symmetry, and Magnetic Uniqueness}},
author = {Dipesh Bhandari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04521},
eprint = {2607.04521},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04521},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperbolic Completion of Newton's Off-Center Orbit Problem: Constant-Curvature Rigidity, SO(2,1) Symmetry, and Magnetic Uniqueness
AU - Dipesh Bhandari
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2607.04521
UR - https://arxiv.org/abs/2607.04521
AB - The hyperbolic potential V(r)=−α/((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)=α/((R^2± 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 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 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.
ER -