# Harmonic oscillator and cyclic coordinates

I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows $$H = \frac{1}{2m}(p^2 + m^2\omega^2q^2)$$.

"This form of the hamiltonian, as sum of two squares, suggests a transformation in which $H$ is cyclic in the new coordinates."

I don't see why is that the case why does it suggest a transformation in which $H$ is cyclic?

• is that because we can somehow relate to sin and cos or is there some other reason ? – Illustionisttt. Dec 10 '15 at 5:21

I don't have enough reputation to comment, but since this is a simple homework problem, let me give you a hint. You know that the total energy $H$ is conserved. You've seen that $H$ can be written in the following form:
$$H = a (X^2 + b Y^2)$$
where $a$ and $b$ are some constants and $X,Y$ can be identified with $p$ and $q$. Let's suppose that $b=1$, which can be achieved by rescaling $Y$. For simplicity we can also set $a=1$ for now. Suppose that the energy of your particle is $E$. Then what are the possible coordinates $(X,Y)$ that your particle can have? Which shape does this describe?