Calculating geodesics:

Abstract

Introduction: The article is the product of the research “Connections on Semi-Riemannian Geometry and Christoffel Coefficients – Towards the study of the computational calculation of geodesics”, developed at the Pascual Bravo University Institution in the year 2021. Problem: Based on solutions of the Euler-Lagrange equations, the explicit calculation of geodesics on certain manifolds is possible. However, there are several cases in which it is impossible to continue calculating analytically and we have to resort to a numerical calculation. In this sense, several geometric and dynamic characteristics of geodesics, unexpectedly emerge. Objective: The objective of the research is to calculate geodesics of a Riemannian or semi-Riemannian manifold using SageMath as software to more easily go beyond what intuition provides. Methodology: First, some simple examples of characterizations of geodesics on certain manifolds, based on solutions of the Euler-Lagrange equations, are presented. Then, an ellipsoid is selected as a test subject with which to numerically calculate geodesics, observing how it changes depending on whether it is defined within a Spherical, Triaxial or Mercator coordinate system. Results: With the flexibility of software like SageMath, an explicit expression of the differential equations was made possible along with, from numeric solutions for these equations, their corresponding simulations depending on the selected parameters. Conclusion: These simulations confirm that great circles are not the only geodesics existing on the ellipsoid, but rather there are many other types of geodesic curves, some of which can be dense curves on the surface and others can be closed curves. At the same time, this shows a relationship between the existence of certain types of geodesic curves and the parameterization of the surface.