An exercise from Goldstein (9.31-3rd Ed) asks to show that for a one-dimensional harmonic oscillator $u(q,p,t)$ is a constant of motion where
$$
u(q,p,t)=\ln(p+im\omega q)-i\omega t
$$
and $\omega=(k/m)^{1/2}$. The demonstration is easy but the physical significance of the constant of motion is not so clear to me. Indeed I can show that $u$ can be rewritten like:
$$
u(q,p,t)=i\phi+\ln(m\omega A)
$$
where $\phi$ is the phase and $A$ the amplitude of the vibration of the oscillator. I can also demonstrate that $m\omega A=\sqrt{2mE}$, where $E$ is the total energy of the oscillator. But there is any further significance of $u$ that I'm missing?