Charged Particle in Magnetic Field

[Spinny]

We have a particle with electric charge e that moves in a strong magnetic field B. The particle is constrained to the (x,y)-plane, and the magnetic field is orthogonal to the plane, and constant with B as the z-component of \mathbf{B}.

Furthermore we have the rotationally symmetric form of the vector potential,

\mathbf{A} = -\frac{1}{2}\mathbf{r}\times \mathbf{B}

the relation between between velovity and momentum

\mathbf{v} = \frac{1}{m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A} \right)

and the Hamiltonian

H = \frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\mathbf{A} \right)^2

We now have to show by use of the equation of motion that, generally, the mechanical angular momentum

L_{mek} = m(xv_y-yv_x)

is not a constant ot motion, whereas

L_{mek}+\left(\frac{eB}{2c} \right)r^2

is conserved.

I'm really not sure where or how to start. Do I have to use the Lagrangian and the Euler-Lagrange equation? If so, how do I find out wether or not any of these expressions are constants of motion or not? Or am I way off?

Edit: minor correction to equations.

[StatusX]

I don't know what you mean in your first equation by taking the cross product of 1/2 and B, but to answer your question, generally conserved quantities arise when the langrangian (or hamiltonian) is symmetric with respect to a change with one of the coordinates. Conservation of angular momentum comes from symmetry in rotations, ie, in the coordinate theta. Find a system of coordinates such that there is some theta-like coordinate with respect to which the lagrangian is invariant.

[Spinny]

I don't know what you mean in your first equation by taking the cross product of 1/2 and B, but to answer your question, generally conserved quantities arise when the langrangian (or hamiltonian) is symmetric with respect to a change with one of the coordinates. Conservation of angular momentum comes from symmetry in rotations, ie, in the coordinate theta. Find a system of coordinates such that there is some theta-like coordinate with respect to which the lagrangian is invariant.

First off, I've corrected the first equation, now it hopefully makes more sense :smile:

Second, I'm afraid it didn't help me much, the rest of your post. You see, this is really a quantum mechanical course, so I'm a bit surprised we got a problem like this. Anyway, I don't know how to find "a system of coordinates such that there is some theta-like coordinate with respect to which the lagrangian is invariant", or even where to begin with all this. I don't think this is supposed to be such a difficult problem, but then again, everything is relative :wink: