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Action-Angle Representation

In mechanics, periodicity, one of the most pervasive aspects of nature, leads to the introduction of action-angle variables as phase space coordinates for a dynamical system. One of the chief virtues of these variables is that they decompose the given dynamical system into its fundamental components, a set of noninteracting subsystems (``degrees of freedom'') each having its own frequency. This is why some mathematicians2, call these coordinates ``normal coordinates'', in analogy with the normal modes of a linear vibrating system.

Suppose the charged particle moves in the periodic electromagnetic field of a plane wave

$\displaystyle A_x(u)$ $\displaystyle = -\frac{E_x}{\omega_1} \sin\omega_1 u$ (30)
$\displaystyle A_y(u)$ $\displaystyle = -\frac{E_y}{\omega_1} \sin(\omega_1 u+\delta)~,$ (31)

which propagates along the positive $ z$-direction with frequency be $ \omega_1$. Then the vector potential, Eq.(13), satisfies

$\displaystyle A_\mu(u+2\pi /\omega_1)=A_\mu (u)~,$ (32)

and so does the superhamiltonian, Eq.(2).



Subsections
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Ulrich Gerlach 2005-11-07