The two-sphere (of radius $R$) can be defined by the following equation in ${\mathbb R}^3$:

(1)We can solve the equation by working in spherical polar coordinates:

(2)where $\theta\in[0,\pi]$ and $\varphi\in [0,2\pi)$.The induced metric on the two-sphere, $S^2$, can be easily shown to be

(3)in spherical polar coordinates. We set $R=1$ for the rest of the discussion.

## Geodesics on the two-sphere

Geodesics are the analog of straight lines in ${\mathbb R}^n$ — they are curves corresponding to the shortest length between any two points. It will be shown that these curves *necessarily* lie on great circles^{1}. Let us first write out the equation of a great circle in spherical polar coordinates. A great circle can be obtained by intersecting a plane passing through the origin in ${\mathbb R}^3$ with the two-sphere. The equation of such a plane is

for some constants $a,\ b,\ c$. A small amount of algebra converts this into the following equation in spherical polar coordinates:

(5)where $A$ and $\varphi_0$ are constants.

### The action for the geodesic

Consider the following action for a particle:

(6)This action has an unusual symmetry i.e., the reparametrisation of ‘time’. Let us write a change of variable given by $t=t(\xi)$. Then, it is easy to see that

(7)where the prime indicates differentiation w.r.t. to the variable $\xi$. In other words, one has the freedom to redefine the time variable. Let us, for instance, choose to parametrise time by the azimuthal angle $\varphi$. Then, one has

(8)where now we consider $\theta(\varphi)$ and the prime indicates differentiation w.r.t. $\varphi$. Thus, the system now effectively has one degree of freedom. The Euler-Lagrange equation of motion is given by

(9)This can be rewritten as

(10)It is now easy to see that $\cot \theta = A\ \cos (\varphi + \varphi_0)$ is indeed the most general solution to the Euler-Lagrange equation. $\varphi$ plays the role of the affine parameter.

A second and more elegant parametrization is to chose the proper length $ds$ as the time coordinate. One then has

(11)Using

(12)The Euler-Lagrange equation for $\theta$ is

(13)can be rewritten as

(14)Similarly, the Euler-Lagrange equation for $\varphi$ is

(15)which can be written as

(16)### Remarks

The form of the Euler-Lagrange equations that appear when we choose the affine parameter as "time" leads to what is known as the geodesic equation on a space with coordinates $x^\mu$:

(17)where $\Gamma^{\mu}_{\nu\rho} =\Gamma^{\mu}_{\rho\nu}$ is the Christoffel symbol/connection. We see that for a two sphere, with $x^\mu=(\theta,\varphi)$, $\Gamma^\theta_{\varphi\varphi}=- \sin\theta\cos\theta$ and $\Gamma^\varphi_{\theta\varphi}=\cot\theta$ are the only two non-vanishing Christoffel symbols.

We can also understand the use of the term *affine* parameter. An affine transformation of a parameter $s$ is the change of variable $s\rightarrow s'=as+b$. It is a simple **exercise** to show that the (form of the) geodesic equation, Eq.(17), is unchanged under affine transformations. Any other transformation will generate extra terms.

where ${x'}^{\mu}(s'):=x^\mu(s)$. Thus the affine parameter for a geodesic is *unique* up to an affine change of variables.