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.