发表机构
Southern Methodist University（南方卫理公会大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
解决奇异势下双曲偏心轨道问题，用雅可比度量分类零能量轨迹，明确朗之万型矩映射，研究反演对偶性及量子力学情况，还涉及磁场分类，数值积分验证相关方程和守恒量。
AI 中文摘要
我们解决了奇异势\[ V(r)=-\frac{\alpha}{(R^2 - r^2)^2},\qquad \alpha>0 \]的双曲偏心轨道问题。在零能量时，雅可比度量在由\(r = R\)分隔的两个分量上具有恒定负曲率。内部雅可比度量是庞加莱圆盘度量的常数倍，而圆反演将外部等距映射到穿孔圆盘。我们对所有零能量轨迹进行分类……最后，保对称的径向磁场在双曲平面上变为恒定固有场。其平移卡西米尔将轨迹分类为闭合磁圆、极限圆或开放超圆，零场测地线为极限情况且在\(Q^2 = 8m\alpha R^2\)处有转变。数值积分证实了轨道方程和守恒量。
英文摘要
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.
Comments22 pages, 3 figures
