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Trajectory in magnetic undulator

  1. May 2, 2015 #1
    1. The problem statement, all variables and given/known data

    I am asked to find r(t) for a charged q particle in an magnetic undulator. Wrote down these equations:
    x'' = w* y' *cos(a*x) (1)
    y'' = -w* x' *cos(a*x) (2)
    z'' = 0 (3)

    r(0) = (x0, y0, z0); r(0)' = (x0', y0', z0').
    2. Relevant equations
    Not sure how to go on solving these.

    3. The attempt at a solution
    z(t) is obvious. I am able to integrate (2) once to find y' = -w/a *sin(a*x) + C. Plugging it into (1) doesn't seem to do any progress, since I get x'' = - w^2 /a *sin(a*x)*cos(a*x) + C*w*cos(a*x). Because particle is in magnetic field, it's known that sqrt(x'^2 + y'^2 + z'^2) = constant from r(0)', but not sure how to use it to my advantage.
     
  2. jcsd
  3. May 2, 2015 #2

    mfb

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    2016 Award

    Staff: Mentor

    A typical undulator would allow some approximations that make the equations easier (e. g. "x' does not change much").
    A fourier transformation could give interesting results (the path is like a sine-curve or a bit similar to a circle, but certainly periodic), but I don't know if it works.
     
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