Characterization of Geodesics

James Holmes

Abstract

Boas defines a geodesic as “the curve along the surface which marks the shortest distance between two neighboring points.”[1]  The term "geodesic" comes from geodesy, the science of measuring the size and shape of the earth.[2] Geodesics are often seen in the study of Riemannian geometry and metric geometry. In physics, more specifically in the theory of general relativity, geodesics can be used to describe a variety of situations, ranging from the path of a falling rock to a planetary orbit. [3]

This project involved using the Euler equation, which is a useful tool for finding extremal functions, to characterize geodesics. After showing the derivation of the Euler equation, it was used to characterize geodesics of a plane, sphere, cylinder, and cone. Then applications of geodesics were investigated.

References
1) Boas, Mary L., Mathematical Methods in the Physical Sciences third edition, Wiley, year, pp. 472-495.
2) Wikipedia, Geodesic, Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, http://en.wikipedia.org/wiki/Geodesic, [09/01/06].
3) Wikipedia, Great Circle, Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, http://en.wikipedia.org/wiki/Great_circle, [09/01/06].

Biography

Research Office, Museum Bldg. 211
406-496-4102
Questions and Information