Simplification of a System of Geodesic Equations by Reference to Conservation Laws

TL;DR

This paper demonstrates how conservation laws, via Noether's theorem, can simplify geodesic differential equations by identifying symmetries, with explicit calculations for geodesics on a glome.

Contribution

It explicitly computes infinitesimal symmetries of geodesic equations using conservation laws, illustrating a method to simplify complex differential equations.

Findings

Explicit symmetries for geodesic equations on a glome identified

Conservation laws facilitate the reduction of differential equations

Method demonstrates potential for broader application in geometric analysis

Abstract

This paper is purposed to exploit prevalent premises for determining analytical solutions to differential equations formulated from the calculus of variations. we realize this premises from the statement of Emmy Noether's theorem; that every system in which a conservative law is observed also admits a symmetry of invariance. As an illustration, the infinitesimal symmetries for Ordinary Differential Equations (O.D.E's) of geodesics of the glome are explicitly computed and engaged following identification of a relevant conservation law in action. Further prospects for analysis of this concept over the same manifold are then presented summarily in conclusion.