5.10: Geodesic
- Page ID
- 14048
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The geodesic is defined as the shortest path between two fixed points for motion that is constrained to lie on a surface. Variational calculus provides a powerful approach for determining the equations of motion constrained to follow a geodesic.
The use of variational calculus is illustrated by considering the geodesic constrained to follow the surface of a sphere of radius \(R\). As discussed in appendix \(19.3.2C\), the element of path length on the surface of the sphere is given in spherical coordinates as \(ds=R \sqrt{d\theta ^{2}+\left( \sin \theta d\phi \right) ^{2}}\). Therefore the distance \(s\) between two points \(1\) and \(2\) is
\[s=R\int_{1}^{2}\left[ \sqrt{\left( \frac{d\theta }{d\phi }\right) ^{2}+\sin ^{2}\theta }\right] d\phi\]
The function \(f\) for ensuring that \(s\) be an extremum value uses
\[f=\sqrt{\theta ^{\prime 2}+\sin ^{2}\theta }\]
where \(\theta ^{\prime }=\frac{d\theta }{d\phi }.\) This is a case where \( \frac{\partial f}{\partial \phi }=0\) and thus the integral form of Euler’s equation can be used leading to the result that
\[\sqrt{\theta ^{\prime 2}+\sin ^{2}\theta }-\theta ^{\prime }\frac{\partial }{ \partial \theta ^{\prime }}\sqrt{\theta ^{\prime 2}+\sin ^{2}\theta }=\text{ constant}=a\]
This gives that
\[\sin ^{2}\theta =a\sqrt{\theta ^{\prime 2}+\sin ^{2}\theta }\]
This can be rewritten as
\[\frac{d\phi }{d\theta }=\frac{1}{\theta ^{\prime }}=\frac{a\csc ^{2}\theta }{ \sqrt{1-a^{2}\csc ^{2}\theta }}\]
Solving for \(\phi\) gives
\[\phi =\sin ^{-1}\left( \frac{\cot \theta }{\beta }\right) +\alpha\]
where
\[\beta \equiv \frac{1-a^{2}}{a^{2}}\]
That is
\[\cot \theta =\beta \sin \left( \phi -\alpha \right)\]
Expanding the sine and cotangent gives
\[\left( \beta \cos \alpha \right) R\sin \theta \sin \phi -\left( \beta \sin \alpha \right) R\sin \theta \cos \phi =R\cos \theta\]
Since the brackets are constants, this can be written as
\[A\left( R\sin \theta \sin \phi \right) -B\left( R\sin \theta \cos \phi \right) =\left( R\cos \theta \right)\]
The terms in the brackets are just expressions for the rectangular coordinates \(x,y,z.\) That is, \[Ay-Bx=z\]
This is the equation of a plane passing through the center of the sphere. Thus the geodesic on a sphere is the path where a plane through the center intersects the sphere as well as the initial and final locations. This geodesic is called a great circle. Euler’s equation gives both the maximum and minimum extremum path lengths for motion on this great circle.
Chapter \(17\) discusses the geodesic in the four-dimensional space-time coordinates that underlie the General Theory of Relativity. As a consequence, the use of the calculus of variations to determine the equations of motion for geodesics plays a pivotal role in the General Theory of Relativity.