The Lagrange Top: Formulas and Analysis for Non-Zero Angular Momentum Cases

[zwierz]

All the needed formulas are here http://hepweb.ucsd.edu/ph110b/110b_notes/node36.html
I consider the following case
$$p_\psi\ne 0,\quad p_\phi/p_\psi\in (\cos\theta_2,\cos\theta_1)$$ this case corresponds to the middle picture in the bottom of the cited page.

I can not prove that the time average of the angle $\phi$ is not equal to zero: $\int_0^\tau\phi(t)dt\ne 0$, here $\tau$ is the period of the function $\theta(t)$.
I know it looks like a standard simple thing but I have been thinking for three days and the result is zero, I also can not find it in books. Please help.

 

[Charles Link]

Very interesting problem. I think I have a qualitative type proof, but it doesn't use the Lagrange equations. The top has a torque on it from gravity. This torque when integrated over time from $0$ to $\tau$ will be non-zero. (The $\vec{r}$ and the $\vec{F}$ don't do a cycle around the top during the time interval of one loop=they basically remain in the same general vicinity for the single loop.) This non-zero result means that the integral of $\frac{d \vec{L}}{dt}$ ($\vec{L}$ is the angular momentum, basically from the spinning top) must be non-zero so that $\Delta \vec{L}$ is non-zero. The top can not return to the same location after doing a small loop, because this would imply $\Delta \vec{L}=0$.

 

[zwierz]

yes but the vector $\boldsymbol L$ is not parallel to the axis $z_B$

UPD: the vectors $\boldsymbol L(t)$ and $\boldsymbol L(t+\tau)$ can have the same direction but different value