5.10: Geodesic

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.