Equations of motion describing a great circle

Question

I'd like to argue that equations of motions of the form $$\ddot \varphi = 0 \quad \text{and} \quad \ddot\theta = \sin\theta\cos\theta\dot\varphi^2$$ describe a great circle.

I think the standard argument goes something like this: $$\ddot\varphi =0\quad \Longrightarrow\quad \dot\varphi = const. =:\omega\quad \Longrightarrow\quad \varphi(t)=\omega t + \varphi_0.$$ We can now fix an initial condition for $\theta$, lets say, $\theta(t=0)=\pi/2$. With this we get $$\ddot \theta(t=0)= 0\quad \Longrightarrow \quad \theta(t)=\frac{\pi}{2},\quad \forall t,$$ which would describe a great circle.

And this last implication is where I get lost. What exactly is the argument that guarantees here that $\theta$ is constant in all time? It seems to be related to $\dot\varphi = const.$ but I just cann't formulate a satisfying argument why $\dot \theta = 0$.

Answer 1

Note that we can perform a fairly standard trick with the $\theta$ equation:

$$2\dot \theta \ddot \theta = 2\sin(\theta)\cos(\theta)\dot \theta \dot \varphi^2$$ $$\implies \frac{d}{dt}\left(\dot \theta ^2\right) = \frac{d}{dt}\left(\dot \varphi^2\sin^2(\theta)\right)$$

since $\dot\varphi$ is constant. Therefore we have that $$\dot\theta^2 = \dot\varphi^2\sin^2(\theta)+C$$ $$\implies \dot\theta = \pm \sqrt{ \dot\varphi^2\sin^2(\theta)+C}$$

This is a bit easier to work with. Note also that you must fix two initial conditions for $\theta$, not just one.