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

TL;DR

This paper uses physics-inspired methods to analyze geodesics on surfaces of revolution, including the torus, and extends the approach to Kepler problems, providing conceptual clarity and insights into closed geodesics and energy quantization.

Contribution

It introduces a physics-based perspective to study geodesics on surfaces of revolution and extends the method to nonrelativistic and relativistic Kepler problems, offering new insights.

Findings

Effective potential approach simplifies geodesic analysis.

Spectrum of closed geodesics resembles atomic energy quantization.

Method bridges geometric and physical interpretations.

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.