Hamiltonian for classical harmonic oscillator

[Jacksond]

I am working through Leonard Susskinds 'the theoretical minimum' and one of the exercises is to show that H=ω/2(p^2+q^2).

The given equations are H=1/2mq(dot)^2 + k/2q^2, mq(dot)=p and ω^2=k/m.

q is a generalisation of the space variable x, and (dot) is the time derivative if this helps. The solution I am getting contains variables in front of the q and p's inside the brackets, do these reduce somehow? Any proof/explanation would be much appreciated :)

 

[WannabeNewton]

So $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 q^2$ is the standard form of the harmonic oscillator Hamiltonian. Now all you have to do is rescale $p$ and $q$ appropriately to get it into the desired form (this rescaling of the conjugate variables is a special case of what are known as canonical transformations).