Action-Angle Variables
We want to find out the periods of periodic trajectories, even for complex trajectories. Especially, if we don’t need to solve the trajectory.
Suppose we have \(H(q,p) = E = \alpha\) is a constant for one dimension. Then, we can get \(p = p(q,\alpha)\). Plotting this orbit in the phase space (q,p), we can get a couple different types of trajectories,
- Liberation/ Oscillation motion: ’circular’ phase space trajectories (as q changes [and repeats itself], p repeats itself).
- Rotation: ’sinusoidal’ phase space trajectories. (as q continues to increase, p repeats itself)
Let’s introduce the action variable \(J\). Let \(J = \oint pdq = \oint p(q,\alpha)dq\) be the integration for one period. Note that \(J\) will no longer be dependent upon \(q\) so the integration remove the dependence of \(q\). So, \(J = J(\alpha)\). Then, we can write \(H = H(J)\). HPF: \(S(q,\alpha,t) = W(q,\alpha) - \alpha t\). So, \(S(q,J,t) = W(q,J) - \alpha(J)t\).
Let \(Q = \beta = \frac{\partial S}{\partial\alpha}\). So, \(S = S(q,\alpha,t) = W(q,\alpha) - \alpha t = W(q,J) - \alpha(J)t\) hence \(Q = \beta = \frac{\partial W}{\partial\alpha} - t\). Also, \(Q = \beta = \frac{\partial W}{\partial J}\frac{\partial J}{\partial\alpha} - \frac{\partial J}{\partial\alpha}t\). Let \(\overline{Q} = \overline{\beta} = \frac{\partial W}{\partial J} - \frac{\partial\alpha}{\partial J}t\). So, \(\overline{Q} = \overline{\beta} = \omega(q,J) - \nu(J)t\). Hence, \(\omega(q,J) = \overline{\beta} + \nu(J)t\).
Under a single period of motion, \(\Delta\omega(q,J) = \oint dq\frac{\partial \omega}{\partial q} = \oint dq\frac{\partial}{\partial q}\left(\frac{\partial W}{\partial J}\right)dq = \frac{\partial}{\partial J}\oint dq\frac{\partial W}{\partial q} = \frac{\partial}{\partial J}\oint dqp = \frac{\partial}{\partial J}J = 1 = \nu(J)T\), where \(T\) is the time of one period. Thus, \(\nu(J) = \frac{1}{T} = \frac{\partial\alpha}{\partial J} = \frac{\partial H}{\partial J}\). Then, for the particular coordinate (for a seperable system), the frequency for the \(\sigma\) coordinate is \(\frac{\partial H(J_1,\cdots,J_n)}{\partial J_\sigma}\).
Example
Suppose we have a one dimensional harmonic oscillator. Then,
\begin{align*} H = \frac{1}{2m}\left(p^2 + m^2\omega^2q^2\right) = \alpha, \omega=\sqrt{\frac{k}{m}} \end{align*}Then,
\begin{align*} J &= \oint dq p \\ &= \pm\oint dq \sqrt{2m\alpha - m^2\omega^2q^2} \\ &= \pm\sqrt{2m\alpha}\oint dq \sqrt{1 - \frac{m^2\omega^2q^2}{2m\alpha}} \\ &= \pm\sqrt{2m\alpha}\oint dq \sqrt{1 - \frac{m^2\omega^2q^2}{2m\alpha}} \\ &= \pm\sqrt{2m\alpha}\sqrt{\frac{2\alpha}{m\omega^2}}\oint d\theta \cos^2\theta \\ &= \frac{2\alpha}{\omega}\int_0^{2\pi}d\theta\cos^2\theta \\ &= \frac{2\alpha}{\omega}\pi. \\ \alpha = \frac{J\omega}{2\pi}. \\ \frac{\partial H}{\partial J} &= \frac{\omega}{2\pi}. \\ T &= \frac{2\pi}{\omega}. \end{align*}\(\frac{m\omega^2}{2\alpha}q^2 = \sin^2\theta\). \(q = \sqrt{\frac{2\alpha}{m\omega^2}}\sin\theta\). \(dq = \sqrt{\frac{2\alpha}{m\omega^2}}\cos\theta d\theta\).