Constructing a Space from the System of Geodesic Equations

TL;DR

This paper presents a method to reconstruct a space's metric from its geodesic equations and extends it to identify when a second order system can be expressed as geodesic equations, supported by a computer implementation.

Contribution

It introduces a procedure to derive the metric tensor from Christoffel symbols and extends it to analyze quadratic systems, including a computer code for larger systems.

Findings

Successfully reconstructs metric tensor from Christoffel symbols.

Provides criteria to identify geodesic systems among quadratic equations.

Develops a computer tool for analyzing complex geodesic systems.

Abstract

Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from the Christoffel symbols. The procedure is extended for determining if a second order quadratically semi-linear system can be expressed as a system of geodesic equations, provided it has terms only quadratic in the first derivative apart from the second derivative term. A computer code has been developed for dealing with larger systems of geodesic equations.