Stationary Position of a 3D Harmonic Oscillator in a constant EM field

[maverick280857]

Hi,

I have to find the 'stationary position' of a particle of mass m and charge q which moves in an isotropic 3D harmonic oscillator with natural frequency \omega_{0}, in a region containing a uniform electric field \boldsymbol{E} = E_{0}\hat{x} and a uniform magnetic field \boldsymbol{B} = B_{0}\hat{z}.

The nonrelativistic Lagrangian of the system is

L = \frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2) - \frac{1}{2}m\omega^2(x^2+y^2+z^2) + qE_{0}x

and the equations of motion are

\ddot{x}+\omega_{0}^2 x - \frac{qB_{0}}{mc}\dot{y} = \frac{qE_{0}}{m}
\ddot{y}+\omega_{0}^2 y + \frac{qB_{0}}{mc}\dot{x} = 0
\ddot{z}+\omega_{0}^2 z = 0

What does "stationary position" mean here? Is it the point where \ddot{x} = \ddot{y} = \ddot{z} = 0? The next part of the question asks to find the equations of motions for oscillations about this position, and the normal modes.

Thanks in advance.

 

[Meir Achuz]

"stationary position" means the position for which all time derivatives are zero.
This means x_0=q E_0/m\omega_0^2.

 

[maverick280857]

"stationary position" means the position for which all time derivatives are zero.
This means x_0=q E_0/m\omega_0^2.

Thanks clem!