17.6: Normal Coordinates

Landau writes \(Q_{\alpha}=\operatorname{Re} C_{\alpha} e^{i \omega_{\alpha} t}\). (Actually he brings in an intermediate variable \(\Theta_{\alpha}\), but we'll skip that.) These “normal coordinates” can have any amplitude and phase, but oscillate at a single frequency \(\ddot{Q}_{\alpha}=-\omega_{\alpha}^{2} Q_{\alpha}\).

The components of the above vector equation read:

\begin{equation}
\begin{array}{l}
\theta_{1}=Q_{1} / \sqrt{3}+Q_{2} / \sqrt{2}+Q_{3} / \sqrt{6} \\
\theta_{2}=Q_{1} / \sqrt{3}-2 Q_{3} / \sqrt{6} \\
\theta_{3}=Q_{1} / \sqrt{3}-Q_{2} / \sqrt{2}+Q_{3} / \sqrt{6}
\end{array}
\end{equation}

It’s worth going through the exercise of writing the Lagrangian in terms of the normal coordinates:

recall the Lagrangian:

\begin{equation}
L=\dfrac{1}{2} \dot{\theta}_{1}^{2}+\dfrac{1}{2} \dot{\theta}_{2}^{2}+\dfrac{1}{2} \dot{\theta}_{3}^{2}-\dfrac{1}{2} \omega_{0}^{2} \theta_{1}^{2}-\dfrac{1}{2} \omega_{0}^{2} \theta_{2}^{2}-\dfrac{1}{2} \omega_{0}^{2} \theta_{3}^{2}-\dfrac{1}{2} k\left(\theta_{1}-\theta_{2}\right)^{2}-\dfrac{1}{2} k\left(\theta_{3}-\theta_{2}\right)^{2}
\end{equation}

Putting in the above expressions for the \(\theta_{\alpha}\), after some algebra

\begin{equation}
L=\dfrac{1}{2}\left[\dot{Q}_{1}^{2}-\omega_{0}^{2} Q_{1}^{2}\right]+\dfrac{1}{2}\left[\dot{Q}_{2}^{2}-\left(\omega_{0}^{2}+k\right) Q_{2}^{2}\right]+\dfrac{1}{2}\left[\dot{Q}_{3}^{2}-\left(\omega_{0}^{2}+3 k\right) Q_{3}^{2}\right]
\end{equation}

We’ve achieved a separation of variables. The Lagrangian is manifestly a sum of three simple harmonic oscillators, which can have independent amplitudes and phases. Incidentally, this directly leads to the action angle variables -- recall that for a simple harmonic oscillator the action \(I=E / \omega\), and the angle is that of rotation in phase space.