Exercise 2

The Hamiltonian is as follows: \[H=p\dot{q}-L\] \[p=\frac{\partial L}{\partial \dot{q}} = \frac{1}{\omega} \dot{q}\] \[p^2 = \frac{1}{\omega^2} \dot{q}^2\] \[H=\frac{1}{\omega} \dot{q} \dot{q} - \frac{1}{2 \omega} \dot{q}^2 + \frac{\omega}{2} q^2\] \[=\frac{\omega}{2}(p^2 + q^2)\]