Given this information, \begin{aligned} \dot\theta &= w \\ \dot{w} &= -\sin(\theta) \end{aligned}
a) Use newton's law to show this describes the dynamics of a pendulum of length $L$, where $g$ is the acceleration of gravity, $\theta$ is the pendulum angle and $w=\dot{\theta}$.
b) Find the function that is invariant on trajectories in state space.
For part a) this is what I have so far..
$\begin{bmatrix} \dot{\theta} \\ \dot{w} \end{bmatrix}=\begin{bmatrix} 0 && 1 \\ -g/L && 0 \end{bmatrix}\begin{bmatrix} \theta \\ w \end{bmatrix}$
So my question would be is part a) correct and how do you do part b). If part a) is wrong, could you please show me the correct state space representation of it. Also please show me how to do part b), that part is just confusing for me... I'm thinking I should consider the energy of the system but I'm not sure.
Thank you