The equation of motions due to the dipole magnetic force of a planet in a frame corotating with the planet and origin at the centre of planet assumed to be sphere components wise are given as below: \begin{alignat}1 \dot r& = x & \\ \dot x &= r[y^2 +(z +Ω)^2(\sin\theta)^2 - \beta z\sin\theta B_\theta)] & \\ \dot \theta &= y & \\ \dot y &= \frac{1}{r}[- 2xy - r(z+ Ω)^2 \sin\theta \cos\theta + \beta r z \sin\theta B_r] & \\ \dot \theta &=z & \\ \dot z &= \frac{1}{r \sin\theta}[-2x(z+Ω)\sin\theta - 2ry(z+ Ω)\cos\theta + \beta(xB_\theta - ryB_r)]& \end{alignat}

The magnetic field components are :
\begin{alignat}1 B_\theta = (\mu_0/4\pi)(R/r)^3 g_{10}sin(\theta)& \end{alignat} \begin{alignat}2 B_r = (\mu_0\theta_0/4\pi)2(R/r)^3g_{10}cos(\theta)& \end{alignat}. The azimuthal magntic component assumed to be zero.

If, the magnetic field is now displaced along the north by say 'a' meter, what would be the new equation of motion? If the whole co-ordinate were to be shifted,it is easy to see that the only change will be the radial components which can be The new radial coordinate (r') will be r' = r - a. But, I dont know how to incorporate the effect magnetic field (Lorent force) in the shifted spherical polar co-ordinate which is also co-rotating with planet and the old/orignal spherical polar coordinates with origin centre of the planet? The particle is the old frame of reference. Thank you



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