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Homework Help: Perturbation Theory - Poincare Method

  1. Jul 10, 2008 #1

    I have some trouble while trying to use the Poincare method in a free fall problem.
    There's some point on earth, that the vector R0 points at. from this point there is an orthonormal coordinate system, and some point of mass at (Rx, Ry, Rz).
    I found the expression for the sum of Coriolis force, the gravity force and the centrifugal force. Though, in this expression, the expression of every component includes a mixture of Rx/y/z, so I can't just say that it's a function of R, and say that R=[tex]\epsilon[/tex]R1+[tex]\epsilon^2[/tex]R2+[tex]\epsilon^3[/tex]R3+...
    and substitute it in the expression of the force.
    My aim is to express the acceleration, and say that [tex]\ddot x = a[/tex], substitute, and compare between the coefficients of the same powers of [tex]\epsilon[/tex].
    Any idea how to do that...? I'm at a loss.


    P.S. the expression I've got for the x component of the acceleration, for example, is:
    [tex]\ddot x = \frac{GM}{\vert \mathbf{R}+\mathbf{R_0 }\vert}\mathbf{R_x} + \frac{\omega^2}{2}\sin{(2\phi)}\mathbf{R_y} -\omega^2\sin^2{(\phi)}\mathbf{R_x}[/tex]+2\omega\dot\mathbf{R_z}\sin{(\phi)}
    (For some reason it doesn't include the last part, outside of the tex tags, if I put it between the tags. I think there's a size limit...)
    Thanks again.
  2. jcsd
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