# 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,

1. Liberation/ Oscillation motion: ’circular’ phase space trajectories (as q changes [and repeats itself], p repeats itself).
2. 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*}

$$\frac{m\omega^2}{2\alpha}q^2 = \sin^2\theta$$. $$q = \sqrt{\frac{2\alpha}{m\omega^2}}\sin\theta$$. $$dq = \sqrt{\frac{2\alpha}{m\omega^2}}\cos\theta d\theta$$.

Created: 2024-05-30 Thu 21:16

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