# 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.

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