Perturbation Theory - Poincare Method

[The-herod]

Hello,

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 (R_x, R_y, R_z).
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 R_x/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.

Thanks!

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.

 

[BigJDubs]

</code>Since you are dealing with a vector equation, it is not possible to directly apply the Poincare method. You need to decompose the equation into components and then apply the Poincare method. For example, take the x-component of the acceleration equation:\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}+2\omega\dot\mathbf{R_z}\sin{(\phi)}Now, you can decompose the equation into R_x, R_y, and R_z components:\ddot x = A_1R_x + A_2R_y + A_3R_zWhere,A_1 = \frac{GM}{\vert \mathbf{R}+\mathbf{R_0 }\vert} - \omega^2\sin^2{(\phi)}A_2 = \frac{\omega^2}{2}\sin{(2\phi)}A_3 = 2\omega\dot\mathbf{R_z}\sin{(\phi)}Now, you can apply the Poincare method to each component separately:\ddot x(R_x) = A_1R_x + \epsilon^2R_x'' + O(\epsilon^3)\ddot x(R_y) = A_2R_y + \epsilon^2R_y'' + O(\epsilon^3)\ddot x(R_z) = A_3R_z + \epsilon^2R_z'' + O(\epsilon^3)Now, you can recombine the components to get the final equation:\ddot x = A_1R_x + A_2R_y + A_3R_z + \epsilon^2(R_x'' + R