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

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