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

Abstract: We resolve the hyperbolic off-center-orbit problem for the singular potential [ V(r)=-\fracα{(R^2-r^2)^2},\qquad α>0. ] At zero energy, the Jacobi metric has constant negative curvature on both components separated by $r=R$. The interior Jacobi metric is a constant multiple of the Poincaré disk metric, while circular inversion maps the exterior isometrically to the punctured disk. We classify all zero-energy trajectories: nonradial orbits are arcs of Euclidean circles orthogonal to $r=R$, radial trajectories lie on lines through the origin, and the force center lies outside every nonradial supporting circle. An explicit Runge--Lenz-type moment map closes into $\mathfrak{so}(2,1)$, whose Casimir is the hyperbolic geodesic Hamiltonian. The canonical cotangent lift of inversion preserves the symmetry generators and maps the zero-energy flow to itself up to positive time reparametrization. The singular circle is reached in finite Newtonian time but lies at infinite Jacobi distance. Quantum mechanically, we distinguish the Stäckel coupling transform from a genuine unitary equivalence, whose Euclidean representative is a divergence-form operator rather than the naive flat Schrödinger operator. The bottom of the hyperbolic continuum maps to the Hardy/oscillation threshold of the inverse-square boundary model. Finally, the symmetry-preserving radial magnetic field becomes a constant intrinsic field on the hyperbolic plane. Its shifted Casimir classifies the trajectories as closed magnetic circles, horocycles, or open hypercycles, with zero-field geodesics as the limiting case and a transition at $Q<sup>2=8mαR<sup>2$. Numerical integrations confirm the orbit equations and conserved quantities.

• The paper delivers a complete classification of zero-energy trajectories in both interior and exterior domains using a Poincaré disk metric.
• The paper reveals a hidden SO(2,1) symmetry through a Runge–Lenz-type moment map and establishes an exact inversion duality between orbit flows.
• The paper extends its analysis with a magnetic classification that distinguishes magnetic circles, horocycles, and hypercycles, supported by numerical integration.

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

Overview and Main Results

This paper delivers a comprehensive solution to the "hyperbolic" analog of Newton's off-center orbit problem for the singular potential $V(r) = -\alpha/(R^2 - r^2)^2$ with $\alpha > 0$. The principal technical contributions include: (1) a complete classification of zero-energy trajectories within both interior ($r<R$) and exterior ($r>R$) domains separated by the singular circle $r=R$; (2) explicit construction of the dynamical $\mathfrak{so}(2,1)$ symmetry, exhibiting a Runge–Lenz-type moment map whose Casimir is the geodesic Hamiltonian of the corresponding hyperbolic (Poincaré disk) metric; (3) derivation of an exact canonical equivalence of the flows—up to time reparametrization—under inversion in the singular circle; (4) detailed analysis of magnetic extensions, elucidating the transition between hypercycle, horocycle, and magnetic circle trajectories; and (5) clarification of the quantum and spectral relationships, separating Stäckel transforms from true unitary equivalences. The paper also provides thorough numerical confirmation of the analytical results.

Zero-Energy Orbit Classification and Geometric Structure

The configuration space consists of two components: $D_R = \{\mathbf{r}\in\mathbb{R}^2 : r < R\}$ (disk) and $E_R = \{\mathbf{r}\in\mathbb{R}^2 : r > R\}$ (exterior), with the circle $r = R$ excluded due to the potential singularity. For energy $V(r) = -\alpha/(R^2 - r^2)^2$0, the trajectory problem can be recast in terms of the Jacobi metric, which, for this potential, becomes a constant multiple of the Poincaré disk metric on $V(r) = -\alpha/(R^2 - r^2)^2$1 with Gaussian curvature $V(r) = -\alpha/(R^2 - r^2)^2$2.

Every nonradial zero-energy trajectory, whether in $V(r) = -\alpha/(R^2 - r^2)^2$3 or $V(r) = -\alpha/(R^2 - r^2)^2$4, is a Euclidean circle orthogonal to $V(r) = -\alpha/(R^2 - r^2)^2$5; radial trajectories are straight lines through the origin. The explicit orbit equation reads

$V(r) = -\alpha/(R^2 - r^2)^2$6

where $V(r) = -\alpha/(R^2 - r^2)^2$7 is a conserved Runge–Lenz-type vector and $V(r) = -\alpha/(R^2 - r^2)^2$8 is angular momentum.

Figure 1: Exact geometry of a nonradial zero-energy orbit; only the segment within $V(r) = -\alpha/(R^2 - r^2)^2$9 is physical, and the force center $\alpha > 0$0 always lies outside the supporting circle, which meets $\alpha > 0$1 orthogonally.

The key geometric conclusion is that the force center $\alpha > 0$2 does not lie inside the supporting circle, marking a sharp contrast to the "spherical" counterpart of [Olshanii, (Olshanii, 2022)]. The orthogonality condition $\alpha > 0$3 for a supporting circle of center $\alpha > 0$4 and radius $\alpha > 0$5 is satisfied, confirming the full geometric classification.

Dynamical $\alpha > 0$6 Symmetry and Canonical Inversion

The system possesses a hidden $\alpha > 0$7 dynamical symmetry. The explicit symmetry generators—angular momentum $\alpha > 0$8 and the Runge–Lenz-type vector $\alpha > 0$9—close under the canonical Poisson bracket relations

$r<R$0

with a quadratic Casimir proportional to the geodesic Hamiltonian for the hyperbolic metric.

A central structural feature is the duality under inversion in the singular circle, mapping $r<R$1 to transformed phase-space coordinates such that the Hamiltonian rescales as $r<R$2, and both $r<R$3 and $r<R$4 are strictly preserved. This establishes an exact mapping—up to positive time reparametrization—between interior and exterior dynamical flows, with the inversion-mapped system retaining all integrable properties.

Figure 2: Circular inversion maps interior orbit segments to exterior branches, leaving the supporting circles fixed and exchanging trajectory domains, with invariance of all key symmetry generators.

Singular Boundary Phenomenology

Approach to the singular circle at $r<R$5 is fundamentally distinct in Newtonian and hyperbolic/Jacobi metrics. While Newtonian dynamics reaches $r<R$6 in finite time, the Jacobi (hyperbolic) distance diverges as $r<R$7, making $r<R$8 an infinite-length ideal boundary. Exterior radial escape to $r<R$9 is at finite Jacobi distance but requires infinite Newtonian time, identifying $r>R$0 with the puncture corresponding to the disk origin under inversion.

Quantum and Spectral Perspectives

The quantum problem features a Stäckel-type correspondence where the hyperbolic Laplacian eigenvalue problem is mapped to a flat Schrödinger problem with singular potential, but this mapping is not a true unitary equivalence—only a parameter transform. The true Hilbert-space equivalence (via an explicit intertwining transformation) leads to a differential operator with both variable-coefficient and first-order derivative terms, distinct from the naïve flat singular operator. The lower edge of the hyperbolic continuum exactly coincides with the Hardy/oscillation threshold for the boundary inverse-square coupling, but self-adjointness of the flat operator requires a domain choice independent of this threshold.

Magnetic Extension and Trajectory Trichotomy

The magnetic extension replaces canonical momenta with covariant momenta in a radially varying field $r>R$1. The symmetry algebra is augmented with a central term that can be absorbed by shifting $r>R$2; the resulting shifted algebra maintains the $r>R$3 structure.

Crucially, under the symmetry-preserving magnetic field, zero-energy trajectories on the disk are classified as closed magnetic circles, horocycles, or open hypercycles, controlled by the quadratic parameter $r>R$4:

• $r>R$5: Closed magnetic circles
• $r>R$6: Horocycles tangent to $r>R$7
• $r>R$8: Open hypercycles meeting $r>R$9 twice

Figure 3: Exact magnetic circle–horocycle–hypercycle trichotomy for various $r=R$0; only orbit segments within $r=R$1 are physical, and transitions occur at the predicted Casimir thresholds.

This trichotomy precisely mirrors the well-known Landau problem on the hyperbolic plane, but with an explicit Newtonian correspondence and coupling dictionary.

Numerical integration of trajectories and conserved quantities corroborates all analytic claims; orbit equations, symmetry vector invariants, and Casimir values are satisfied to near machine precision.

Figure 4: Numerical integration of zero-energy magnetic orbits with varying $r=R$2, demonstrating exact agreement between direct integration results (solid) and circle predictions (dashed).

Figure 5: Direct comparison of numerical Hamiltonian integration and prediction from conserved quantities for the nonmagnetic orbit; orbits are indistinguishable.

Figure 6: Dimensionless numerical residuals demonstrating conservation of Hamiltonian, angular momentum, Runge–Lenz vector, and the orbit equation throughout the integration interval.

Implications and Perspectives

The solution of the hyperbolic off-center orbit problem furnishes a concrete integrable system combining singular Newtonian dynamics with nontrivial Jacobi metric geometry and a full symmetry structure. The inversion duality and preservation of dynamical invariants under canonical inversion represent a significant advance in the geometric and algebraic understanding of such systems. In the magnetic case, the embedding of the entire hyperbolic Landau classification within a Newtonian problem—complete with explicit Casimir-based regimes and transitions—provides a template for analogous "geometric completions" of other classical and quantum systems with singularities and nontrivial configuration space topology.

The separation of differential equation correspondence (Stäckel transforms) from true spectral and operator equivalence has implications for quantum integrable systems with singular boundaries.

Future work may target the quantum magnetic extension, including quantization of the shifted $r=R$3 algebra and systematic comparison of its Casimir with spectral features of quantum Landau levels on hyperbolic spaces, accounting for operator-ordering and domain subtleties.

Conclusion

The paper achieves the complete hyperbolic classification of Newton's off-center orbit problem for a singular potential, rigorously develops its hidden $r=R$4 dynamical symmetry, and establishes a canonical duality via inversion, both classically and quantum-mechanically. The full characterization of magnetic trajectory regimes explicitly realizes the geometric classification of constant-field flows on the hyperbolic disk within a Newtonian framework, extending both the theoretical toolkit and the landscape of integrable systems with singularities and rich symmetry.