# Charged Particle in Magnetic Field

by Spinny
Tags: charged, field, magnetic, particle
 P: 20 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.