How Does a Magnetic Field Affect Eigenfrequencies in a Harmonic Oscillator?

[qspeechc]

Homework Statement

Hello everyone. I'm trying to find the eiegenfrequencies in a problem.

We have a particle of mass m, charge e, that is subject to a linear restoring force
\textbf{F} = -k\textbf{r}
and the particle is in a magnetic field \textbf{B} = B\textbf{k}[/itex]<br /> (in the z-direction)<br /> <br /> <h2>The Attempt at a Solution</h2><br /> I am a bit confused ecause I only get one eigen-frequency: sqrt(k/m), which is as though the magnetic field is not there.<br /> <br /> In finding the freq.s you determine the kinetic and potential energy. Th magnetic field has no addittion to the potential energy? This is what I did. I took:<br /> U = 0.5k(x^2 + y^2 + z^2)<br /> T = 0.5mv^2<br /> where v = &lt;dx/dt, dy/dt, dz/dt&gt;<br /> which gave me the above answer, which I suspect is wrong, partly because the question goes on to say:<br /> &quot;Write your answer in terms of sqrt(k/m) and the cyclotron freq. Hint: use the variable u=x+iy&quot;<br /> <br /> I would be grateful for any help.

 

[Irid]

Well, electromagnetic Lagrangian is a little different, because there is a velocity-dependent force. I think it is

L = T - q\phi + q \mathbf{v\cdot A}

where A is the vector potential of magnetic field.

 

[qspeechc]

Hhmm. I'll have to look it up. We never did the Lagrangian when considering velocity-dependent forces. I guess I'll have to look in another book. Thanks for the help Irid.

 

[qspeechc]

I'm sorry, this still isn't working. Do I assume each component x, y, z undergoes SHM, ie are of the form cos(wt), where w is the eigenfrequency, and then solve for w?