We have a particle with electric charge [tex]e[/tex] that moves in a strong magnetic field [tex]B[/tex]. The particle is constrained to the (x,y)-plane, and the magnetic field is orthogonal to the plane, and constant with [tex]B[/tex] as the [tex]z[/tex]-component of [tex]\mathbf{B}[/tex].(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]\mathbf{A} = -\frac{1}{2}\mathbf{r}\times \mathbf{B}[/tex]

the relation between between velovity and momentum

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

and the Hamiltonian

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

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

[tex]L_{mek} = m(xv_y-yv_x)[/tex]

isnota constant ot motion, whereas

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

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.

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# Homework Help: Charged Particle in Magnetic Field

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