Abstract
This paper is purposed to exploit prevalent premises for determining analytical solutions to di erential 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 conservation law is observed also admits a symmetry of invariance (Olver, 1993, pp.242; Dresner, 1999, pp.60-62 ). As an illustration, the infinitesimal symmetries for Ordinary Di erential 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.
Highlights
The 3-sphere, otherwise termed the glome, is a Riemannian manifold at the center of several revolutionary conjectures and advancements in modern mathematical theory
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
The requisite techniques harnessed from the calculus of variations in the author’s prior arXiV publication cover issues such as well-posedness and determination of integral curves of the formulated system of non-linear differential equations via suitable coordinate systems
Summary
The 3-sphere, otherwise termed the glome, is a Riemannian manifold at the center of several revolutionary conjectures and advancements in modern mathematical theory. One may consider the famous Poincareconjecture and the Ricci Flow theorem of Hamilton on closed 3-dimensional manifolds with everywhere positive scalar curvature (Cao et al, 2003, pp.128). The former example challenges an interested mind on meticulous details of differential topology, while the latter is a relatively modern sprout of Pseudo-Riemannian geometry requiring assorted topological and analytical tools. The requisite techniques harnessed from the calculus of variations in the author’s prior arXiV publication cover issues such as well-posedness and determination of integral curves of the formulated system of non-linear differential equations via suitable coordinate systems
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