A simple pendulum is modeled by the (nondimensionalized) differential equation $\frac{d^{2}\theta}{dt^2} = -\sin(\theta)$
With the auxiliary variable $w$, this is equivalent to the first-order system $\begin{align*}\frac{d\theta}{dt}&= w\\ \frac{dw}{dt}&= -\sin(\theta)\end{align*}\qquad (3)$
We know that this system has equilibrium at $(\theta, w) = (2n\pi,0)$ and $(\theta,w) = ((2n+1)\pi,0)$. The second set of equilibria were saddle points, and therefore unstable, but the first set were centers whose stability we were able to determine. In this problem, we'll use a conserved quantity of this system to prove that these fixed points are in fact stable.
The quantity $E = (1/2)w^2 - \cos(\theta)$ is proportional to the total energy (kinetic plus potential) of the pendulum. We expect this to be constant on physical grounds, Use $(3)$ to prove that $dE/dt = 0$
I think I need to solve the first differential because I tried to solve the problem without it and got nowhere. I'm not really sure how to solve it though. Any ideas on how to proceed?