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Frequency of Rotation

4. The evolution of a particle in the periodic electromagnetic field of a travelling plane wave is mathematically equivalent to four rotation processes. Together these processes, Eq.([*])-(51), describe a point moving uniformly on a four dimensional torus. The four radii of the torus are proportional to the action of four degrees of freedom and their respective frequencies are

$\displaystyle \nu_u$ $\displaystyle \equiv\frac{d\phi^u}{d\tau}= \frac{\partial H(\mathbf{I})}{\partial I_u}= \left( -2\frac{\omega_2 I_v}{2\pi m}\right) \frac{\omega_1}{2\pi}$ (55)
$\displaystyle \nu_v$ $\displaystyle \equiv\frac{d\phi^v}{d\tau}= \frac{\partial H(\mathbf{I})}{\partial I_v}= \left( -2\frac{\omega_1 I_u}{2\pi m}\right) \frac{\omega_2}{2\pi}$ (56)
$\displaystyle \nu_x$ $\displaystyle \equiv\frac{d\phi^x}{d\tau}= \frac{\partial H(\mathbf{I})}{\partial I_x}= \left( \frac{ I_x}{m L_x}\right) \frac{1}{L_x}$ (57)
$\displaystyle \nu_y$ $\displaystyle \equiv\frac{d\phi^y}{d\tau}= \frac{\partial H(\mathbf{I})}{\partial I_y}= \left( \frac{ I_y}{m L_y}\right) \frac{1}{L_y}~.$ (58)

These equations will be recognized as the first half of Hamilton's equations relative to action-angle variables. The second half are simply

$\displaystyle \frac{dI_u}{d\tau}$ $\displaystyle =-\frac{\partial H((\mathbf{I})}{\partial \phi^u}~(=0)$ (59)
$\displaystyle \frac{dI_v}{d\tau}$ $\displaystyle =-\frac{\partial H((\mathbf{I})}{\partial \phi^v}~(=0)$ (60)
$\displaystyle \frac{dI_x}{d\tau}$ $\displaystyle =-\frac{\partial H((\mathbf{I})}{\partial \phi^x}~(=0)$ (61)
$\displaystyle \frac{dI_y}{d\tau}$ $\displaystyle =-\frac{\partial H((\mathbf{I})}{\partial \phi^y}~(=0)~,$ (62)

and they express what one knew all along, namely that the action variables are constant along the particle path.


next up previous contents
Next: Conclusion Up: Action Variables Previous: Rotational Periodicity   Contents
Ulrich Gerlach 2005-11-07