Exactly the electron motion equation in the ELECTROMAGNETIC field reads:
r''+a r''' + b r = F.
What is the physical meaning of the third derivative?
The third derivative conveys us that the movement of electron in the field will have parabolic path. Probably, Magnetron is the example of such movement of electron
Let 'x' be the position. First derivative indicates displacement (rate of change of x w.r.t time). Then its second derivative (or derivative of displacement) is the acceleration (how fast or slow displacement (or motion) is taking place. When it comes to third derivative (or derivative of acceleration) it indicates the curved or parabolic motion (i.e., how acceleration is changing w.r.t. time). In your case it is the motion of electrons under influence of electromagnetic fields. For details you may pl. ref. text book on microwaves by Samuel Leo.
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