stability of $(0,0)$ for $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$

Question

Given the system $\dot{\theta} = y$ and $\dot{y} = -\sin\theta$.

For the fixed point $(0,0)$ I can see through linearisation that the flow corresponds to a centre which moves anticlockwise

Why is this centre stable? at $y=0$, $\dot{\theta}=0$ but i cant seem to go from there

Answer 1

Notice that $$ \frac{d}{dt}(\frac12 y^2-\cos \theta) = y\dot{y} + \sin\theta\,\dot{\theta} = 0$$ Can you continue?