%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: geodesic
%%% Primary Category Code: 00.
%%% Filename: Geodesic.tex
%%% Version: 18
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% this is the default PlanetPhysics preamble. as your
% almost certainly you want these
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate}
\usepackage{xypic, xspace}
\usepackage[mathscr]{eucal}
\usepackage[dvips]{graphicx}
\usepackage[curve]{xy}
% define commands here
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\grpL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\rO}{{\rm O}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\SL}{{\rm Sl}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\Symb}{{\rm Symb}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
%\newcommand{\grp}{\mathcal G}
\renewcommand{\H}{\mathcal H}
\renewcommand{\cL}{\mathcal L}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathsf{G}}}
\newcommand{\dgrp}{{\mathsf{D}}}
\newcommand{\desp}{{\mathsf{D}^{\rm{es}}}}
\newcommand{\grpeod}{{\rm Geod}}
%\newcommand{\grpeod}{{\rm geod}}
\newcommand{\hgr}{{\mathsf{H}}}
\newcommand{\mgr}{{\mathsf{M}}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathsf{G)}}}
\newcommand{\obgp}{{\rm Ob(\mathsf{G}')}}
\newcommand{\obh}{{\rm Ob(\mathsf{H})}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\grphomotop}{{\rho_2^{\square}}}
\newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\grplob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\grpa}{\grpamma}
%\newcommand{\grpa}{\grpamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\ovset}[1]{\overset {#1}{\ra}}
\newcommand{\ovsetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}
\newcommand{\<}{{\langle}}
\def\baselinestretch{1.1}
\hyphenation{prod-ucts}
%\grpeometry{textwidth= 16 cm, textheight=21 cm}
\newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}&
#3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram
\eqno{\mbox{#9}}$$ }
\def\C{C^{\ast}}
\newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}}
%\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$
%{\mbox{}}
\newcommand{\quadr}[4]
{\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4&
\end{pmatrix}}
\def\D{\mathsf{D}}
\begin{document}
A \emph{geodesic} is generally described as the shortest possible, or topologically allowed, path between two points in a curved space.
\begin{remark}
Given a curved space $\S_C$ one can find the geodesic by writing the equation for the length $l_v$ of a {\em curve}-- which is defined as a \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $f: (R) \to \S_C$ from an open interval $(R)$ of $\R$ to the \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} $\S_C$-- and then by using the calculus of variations minimizing this length. In physical
applications, however, to simplify the calculation one may also require the minimization of \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} as well as the length of the curve.
However, in \htmladdnormallink{Riemannian geometry}{http://planetphysics.us/encyclopedia/NonabelianAlgebraicTopology3.html} geodesics are not coinciding with the ``shortest length curves'' joining two points, even though a close connection may exist between geodesics and the shortest paths; thus, moving around a great circle on a Riemann sphere the `long way round' between two arbitrary, fixed points on a sphere is a geodesic but it is not obviously the shortest length curve between the points (which would be a straight line that is not permitted
by the topology of the surface of the Riemann sphere).
\begin{example}
The orbits of satellites and planets are all geodesics in curved
\htmladdnormallink{spacetime}{http://planetphysics.us/encyclopedia/SR.html}. As a more general physical example in general relativity theory, \emph{relativistic geodesics} describe the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} of \htmladdnormallink{point particles}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} in a spacetime with a curvature determined only by gravity.
Consider such a point particle $ z^{\mu}$ that moves along a trajectory or ``track'' in physical spacetime; also assume that the track is parameterized with the values of $ \tau $. Then, the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} pointing in the direction of motion of the point particle in spacetime can be written as:
$$ u^{\mu} = { dz^{\mu} \over d\tau }. $$
If there are no \htmladdnormallink{forces}{http://planetphysics.us/encyclopedia/Thrust.html} acting on a point particle, then its velocity is unchanged along the trajectory or `track' and one has the following
\emph{geodesic equation}:
$$ { d u^{\nu} \over d \tau} + \Gamma^{\nu}_{\mu \sigma} u^{\mu} u^{\sigma} \quad = \quad { d^2 z^{\nu} \over d \tau^2} + \Gamma^{\nu}_{\mu \sigma} { d z^{\mu} \over d \tau} { d z^{\sigma} \over d \tau} \quad = \quad 0 . $$
\end{example}
\end{remark}
\begin{definition}
More generally, a \emph{geodesic} in \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html} geometry is defined as a
a curve $\Gamma: I \to M$ from an interval $I \subset \R$ to the \htmladdnormallink{metric space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $M$ for which there exists a constant $v \leq 0$ such that for any $t \in I$ there is a neighborhood $J$ of $t \in I$ such that for any $t_1, t_2 \in J$ one has that
$$d(\Gamma(t_1),\Gamma(t_2)) = v|t_1-t_2|.\,$$
\end{definition}
When the equality
$$ d(\Gamma(t_1),\Gamma(t_2))=|t_1-t_2| \, $$
is satisfied for all $t_1, t_2 \in I$, the geodesic is called the shortest path or a {\em minimizing geodesic}.
\end{document}
