(Solved): The equation of a pendulum having an attached mass \( m>0 \), a massless rod of length \( l>0 \) i ...

The equation of a pendulum having an attached mass \( m>0 \), a massless rod of length \( l>0 \) is \[ m l \theta^{\prime \prime}=-m g \sin (\theta), \] where \( \theta(t) \) is the angular position of the pendulum as function of time, measured from the vertical downwards position, positive counter-clockwise, and \( g \) is the acceleration of gravity. We consider the particular case \( \frac{g}{l}=1 \) and \( m=1 \). Part 1: First Order Reduction (a) Find the first order reduction of the equation above where \( u=\theta \) and \( v=\theta^{\prime} \).