Consider the following system of differential equations $$\begin{cases} x' = y \\ y' = -w^2 \sin(x)-ay \end{cases}$$
with $w > 0 $ and $ a \ge 0$. I know that $V(x,y) = \frac{y^2}{2} + w^2(1-\cos x) $ is a strict Ljapunov function.
I am looking for a proof that every solution $(x,y)$ to the initial value $(x_0,y_0) \in \mathbb{R}^2$ is bounded for $ t \ge 0$.
One can show that $y$ is bounded using a proof by contradiction and the properties of the Ljapunov function. Further I know one way to prove that $x$ is bounded but this one is very long and counterintuitive. (cf. Gewöhnliche Differentialgleichungen und dynamischen Systeme written by Prüss/Wilke)
Therefore I ask you to provide another short/beautiful proof of the above claim.
To clarify: I am looking for an rigorous proof and not an physical idea which somehow shows the boundedness of solutions. While these are the differential equations for the damped pendulum. It is nowhere stated that one of this variables is only an angle. So $(x,y)$ are both in $\mathbb{R}^2$ and I want them to be bounded as such.