Solving Rotation Condition for Equation of Motion

[KFC]

If I know the equation of motion of the following form

$$\ddot{\theta} + k^2\sin\theta = 0$$

(for pendulum for example). What's the condition (minimum angular velocity) to keep it rotate instead of just oscillation?

 

[adriank]

Writing $$\omega = \dot \theta$$, that equation becomes (using the chain rule)

$$\omega \frac{d\omega}{d\theta} + k^2 \sin \theta = 0$$

and solving that differential equation gives

$$\frac12 \omega^2 = k^2 \cos \theta + C.$$

Once you have that, what condition on C do you need for $$\theta = \pi$$ to be possible? (Note that each side of this equation corresponds nicely to energy.)