# Does the motion of a free electron in a plane wave electromagnetic field have analytical solutions?

Does the motion $[x(t), y(t), z(t)]$ of a free electron in plane wave electromagnetic field have analytical solution (oscillating E and B fields)?

The Lorentz force is :

\begin{align}m_e\begin{bmatrix}x''\\y''\\z''\end{bmatrix} &= -e\begin{bmatrix}E\\0\\0\end{bmatrix}e^{i\omega t} -e\begin{bmatrix}x'\\y'\\z'\end{bmatrix} \times \begin{bmatrix}0\\B\\0\end{bmatrix}e^{i\omega t} \\ &= -e\left( \begin{bmatrix}E\\0\\0\end{bmatrix}+\begin{bmatrix}-z'B\\0\\x'B\end{bmatrix} \right)e^{i\omega t} \end{align}

• Have you attempted to solve this problem? Are you asking if there is an analytical solution? Or if there is a figure-of-eight solution? – sammy gerbil Jul 9 '16 at 23:31
• Yes I have tried the result that mathematica gives is quite complicated. z=(E*(t - int(cos((Ba0*(sin(p + wx) - sin(p)))/w), x, 0, t)))/B This is from matlab simplification, still not analytical. Its an integral over infinate series of bessel functions. – Anonymous Jul 9 '16 at 23:33
• In a region of uniform and constant electric and magnetic fields, the solution is analytic if you neglect bremstralung, and it is a helix (or some degenerate limit of a helix). – dmckee Jul 10 '16 at 0:09
• Is the figure 8 is a solution to plane wave EM field in certain frame, still oscillating fileds ? – Anonymous Jul 10 '16 at 0:19
• I would assume you could tune the amplitude and frequency of the plane wave to maintain such a state. It's not clear to me how you would set up the initial conditions to realize it in real life, but on the white-board it should be possible. – dmckee Jul 10 '16 at 0:21