Constant of motion 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?
 A: It is a functional combination of other constants: the energy (another constant of motion) and the initial condition. This would be the same as proving that, in classical mechanics, $E^2 + \log(L)$ with $L$ being the total angular momentum is a constant. It does not hold any new physical meaning, further than what you got.
If you didn't know all this, you could use the fact that $u$ is a constant to show that the amplitude or the energy are constants.
A: The quantity inside the natural log seems to be proportional to the classical analog of the raising operator in quantum mechanics:
$$
a_{+}= \frac{1}{\sqrt{2m}}\bigg(\frac{\hbar{}}{i}\frac{d}{dx}+im\omega{}x\bigg)\\
a_{+}= \frac{1}{\sqrt{2m}}\bigg(\hat{p}+im\omega{}q\bigg)
$$
Where $\hat{p}$ is the quantum mechanical momentum operator and I have changed x to the generalized coordinate q to show the similarity to the problem. 
As you noted, $\omega{}t$ is related to $\phi{}$.
Conclusion: This constant of motion, u, is probably related to the raising operator for a time dependent problem.
