Hyperbolic Completion of Newton's Off-Center Orbit Problem: $SO(2,1)$ Symmetry, Inversion Duality, and Magnetic Classification

Abstract

Which central forces produce circular trajectories whose geometric center differs from the force center? We solve the hyperbolic version of this problem for $$ V(r)=-\fracα{(R^2-r^2)^2},\qquad α>0, $$ whose singular circle $r=R$ separates the configuration space into two components. At zero energy, the Jacobi metric is proportional to the Poincaré disk metric. Hence every nonradial orbit is an arc of a Euclidean circle orthogonal to $r=R$, while radial orbits lie on lines through the origin. We construct a Runge--Lenz-type vector which, together with angular momentum, defines an on-shell $\mathfrak{so}(2,1)$ moment map. Circular inversion preserves this structure and relates the exterior and punctured-interior flows up to time reparametrization. Although $(r=R)$ is infinitely distant in the Jacobi metric, it is reached in finite Newtonian time. A magnetic deformation corresponds to a constant intrinsic field on the hyperbolic plane and yields an exact circle--horocycle--hypercycle transition at $Q^2=8mαR^2$, with inversion acting as the charge-reversing duality $Q\leftrightarrow -Q$. We also relate the hyperbolic continuum threshold to the Hardy threshold of an inverse-square boundary model.