I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using cylindrical coordinates (with the axis of the cylinder in the z direction), describe the motion in terms of the corresponding action-angle variables, showing how the frequencies can be obtained. Transform to the "proper" action-angle variables to eliminate degenerate frequencies.
This is my solution:
Representing Lagrangians in a cylindrical coordinate system:
$\begin{align*}&T=\frac{1}{2}m\left(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2\right),\;\,V=\frac{1}{2}k_1 r^2+\frac{1}{2}k_3 z^2\\&\rightarrow\mathcal{L}=T-V=\frac{1}{2}m\left(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2\right)-\frac{1}{2}k_1 r^2-\frac{1}{2}k_3 z^2\end{align*}$
Thus the generalized momenta are
$p_r=\frac{\partial\mathcal{L}}{\partial\dot{r}}=m\dot{r},\;\,p_\theta=\frac{\partial\mathcal{L}}{\partial\dot{\theta}}=mr^2\dot{\theta},\;\,p_z=\frac{\partial\mathcal{L}}{\partial\dot{z}}=m\dot{z}$
And the Hamiltonian is
$\mathcal{H}=p_r\dot{r}+p_\theta\dot{\theta}+p_z\dot{z}-\mathcal{L}=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{p_z^2}{2m}+\frac{1}{2}k_1r^2+\frac{1}{2}k_3z^2=T+V.$
Here $\frac{\partial\mathcal{L}}{\partial \theta}=0$ so $\theta$ is a cyclic coordinate and $p_\theta$ is an action variable itself.
Then I organize the Hamiltonian in terms of the conserved terms $E_r$, $E_\theta$, and $E_z$ as follows:
$\mathcal{H}=\left(\frac{p_r^2}{2m}+\frac{1}{2}k_1r^2\right)+\frac{p_\theta^2}{2mr^2}+\left(\frac{p_z}{2m}+\frac{1}{2}k_3z^2\right)=E_r+E_\theta+E_z$
Now find the action-angle variables for each coordinate. Expressing the expressions for $E_r$ and $E_z$ in the form $\frac{1}{2}(p^2+\omega^2q^2)=E$, and $I=E/\omega$, so we have
$\begin{align*}&\omega_r=\sqrt{\frac{k_1}{m}}\rightarrow I_r=\frac{E_r}{\omega_r}=E_r\sqrt{\frac{m}{k_1}}\\&\omega_z=\sqrt{\frac{k_3}{m}}\rightarrow I_z=\frac{E_z}{\omega_z}=E_z\sqrt{\frac{m}{k_3}}\\&I_\theta=p_\theta\rightarrow\omega_\theta=\frac{E_\theta}{I_\theta}=\frac{p_\theta}{2mr^2}\end{align*}$
Finally the Hamiltonian is represented as
$\mathcal{H}=\omega_rI_r+\omega_\theta I_\theta+\omega_z I_z$
Questions:
- Do I understand the meaning of "eliminate degenerate frequencies" correctly?
- I think $\omega_\theta$ may have the value of $\dot{\theta}$, but in my solution it is $\dot{\theta}/2$. Is it right?
