What exactly are great circles, and how does one derive them? Given that the Lagrangian is: $$ L =\frac {1}{2}(\dot\theta^2 + \sin^2\theta\dot\phi^2)$$ it was written that the great circles were $$\theta = \tau$$ and $$\phi = \phi_0$$
I have no idea how this comes about. Is there some decent procedure to calculate the great circles? I don't want the answer nor do I want you to try to solve it, I just want to be guided in a way to understand how to get them alone.
As Wikipedia says, a great circle is a circle formed by the intersection of a sphere and a plane that passes through the center of the sphere.
The great circles parametrized by $\theta =\tau$ and $\phi =\text{const}$ are not all the great circles. They are only the `vertical' great circles, that is, the great circles formed by intersection with a plane that contains the $z$-axis. (You will find that the general solution to the Euler-Lagrange equations of your Lagrangian is $\theta (t)=at+b$, $\phi =\text{const}$, for some constants $a$ and $b$. We an then just define $\tau :=at+b$, so that $\theta =\tau$. This is nothing more than a simple reparametrization.) If you need help visualizing these curves, I suggest you work on your understanding of spherical coordinates.
Now I will sketch how one actually shows that these great circles are the equations of motion. You should find that the Euler-Lagrange equations are \begin{align} \ddot{\theta} & =\dot{\phi}^2\sin (\theta )\cos (\theta ) \\ \frac{\mathrm{d}}{\mathrm{d}\, t}\left( \sin (\theta )^2\dot{\phi}\right) & =0. \end{align} The latter equation implies that $\sin (\theta )^2\dot{\phi}=\text{const}$. As the particle will at some point be located at one of the poles, we find that this constant is $0$ (by plugging in $\theta =0,\pi$). Thus, $\dot{\phi}=0$. The first equation then becomes $\ddot{\theta}=0$.
