[Paper Review] Geodesics on the Torus and other Surfaces of Revolution Clarified Using Undergraduate Physics Tricks with Bonus: Nonrelativistic and Relativistic Kepler Problems

Robert T. Jantzen|arXiv (Cornell University)|Dec 26, 2012

Relativity and Gravitational Theory7 references5 citations

TL;DR

This paper clarifies geodesics on surfaces of revolution—particularly the torus—by applying undergraduate physics concepts like energy conservation and effective potentials, treating geodesics as trajectories of point particles under fictitious forces. It demonstrates that geodesic behavior, including focusing and uniqueness, mirrors classical mechanics, and extends the method to both nonrelativistic and relativistic Kepler problems with quantitative results on geodesic convergence and minimal path uniqueness.

ABSTRACT

In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by interpreting parametrized geodesics as the paths traced out in time by the motion of a point in the surface, identifying the parameter with the time. Considering energy levels in an effective potential for the reduced motion then proves to be an extremely useful tool in studying the behavior and properties of the geodesics. The same approach can be easily tweaked to extend to both the nonrelativistic and relativistic Kepler problems. The spectrum of closed geodesics on the torus is analogous to the quantization of energy levels in models of atoms.

Motivation & Objective

• To bridge the gap between differential geometry and classical mechanics by recasting geodesic problems as particle motion in effective potentials.
• To provide a physically intuitive, accessible method for analyzing geodesics on surfaces of revolution using conservation laws familiar to physics undergraduates.
• To extend the approach to both nonrelativistic and relativistic Kepler problems, showing the same formalism applies to orbital motion.
• To clarify the conditions under which geodesics are unique and when they focus, using curvature and effective potential analysis.
• To demonstrate that the minimal geodesic between two points on a torus is guaranteed when their Euclidean distance is less than b, the minor radius.

Proposed method

• Interpret geodesics as time-parametrized particle trajectories on a surface, using the parameter as time to apply energy and angular momentum conservation.
• Reduce the 2D geodesic problem to 1D radial motion using an effective potential that includes the centrifugal barrier from angular momentum.
• Use the effective potential diagram to qualitatively analyze geodesic types (e.g., equatorial, periodic, bounded, unbounded).
• Apply the geodesic deviation equation to study convergence of nearby geodesics, linking curvature to focusing behavior.
• Derive the radial half-period L = π/ωₛ and relate it to the Gaussian curvature, showing that L > bπ for non-spherical tori.
• Use computer algebra systems (Maple, Mathematica) to numerically solve boundary value problems, ensuring minimal-length geodesics are found when points are within a sphere of radius b.

Experimental results

Research questions

• RQ1 How can the geodesic problem on a surface of revolution be reinterpreted using physics-based intuition from classical mechanics?
• RQ2 What determines the uniqueness of a geodesic connecting two points on a torus, and how does this relate to the Euclidean distance between them?
• RQ3 How does positive curvature on the outer equator of a torus lead to geodesic focusing, and what is the physical significance of this convergence?
• RQ4 Can the same effective potential method used for geodesics be extended to describe both nonrelativistic and relativistic Kepler problems?
• RQ5 What is the role of the effective potential barrier in determining the types of geodesic motion (e.g., bounded, unbounded) on a torus?

Key findings

• Geodesics on the torus are uniquely determined between two points if their Euclidean distance is less than the minor radius b, ensuring the minimal-length path is found numerically.
• The convergence length L = π/ωₛ exceeds bπ for all non-spherical tori (c > -1), meaning geodesics starting from the outer equator cross within this distance due to positive curvature.
• The maximum geodesic convergence occurs at the outer equator, where the curvature scalar R̂rθrθ = 1/(b²(c+2)) is maximized, leading to the fastest focusing of nearby geodesics.
• For spindle tori, the curvature scalar diverges to -∞ as r → bπ on the inner hemisphere, but remains positive at the inner equator, with a convergence length b(−c)¹ᐟ²π that decreases as c → 0⁻.
• The geodesic deviation equation reduces to d²r/ds² = −ωₛ²r on the outer equator, confirming the same oscillatory behavior as derived from the effective potential.
• The method successfully generalizes to the relativistic Kepler problem, showing that the same effective potential formalism applies to motion in a Schwarzschild-like gravitational field.